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MATHEMATICAL  MONOGRAPHS. 

EDITED    BY 

Mansfield  Merriman  and  Robert  S.  Woodward. 

Octavo,  Cloth,  $i.oo  each. 

No.  1.     HISTORY  OF  MODERN  MATHEMATICS. 

By  David  Eugene  Smith. 

No.  2.    SYNTHETIC  PROJECTIVE   GEOMETRY. 

By  George  IJruce  Halsted. 

No,  3.    DETERMINANTS. 

By  Laenas  Gifford  Weld. 

No.  4.    HYPERBOLIC   FUNCTIONS. 

By  James  McMahon. 

No.  5.    HARMONIC   FUNCTIONS. 

By  William  E.  Bverly. 

No.  6.    GRASSMANN'S  SPACE  ANALYSIS. 

By  Edward  W.  Hyde. 

No.  7.     PROBABILITY    AND  THEORY    OP    ERRORS. 

By  Robert  S.  Woodward. 
No.  8.    VECTOR   ANALYSIS   AND  QUATERNIONS. 

By  Ale.xa.nder  Macfarlane. 

No.  9     DIFFERENTIAL   EQUATIONS. 

By  Willi A.M  Woolsey  Johnson. 
No.  10     THE  SOLUTION  OF  EQUATIONS. 
By  Mansfield  Meuri.man-. 
No.  11.    FUNCTIONS  OF  A   COMPLEX  VARIABLE. 

By  Thomas  S.  Fiske. 

PUBLI.SHED    BY 

JOHN   WILEY  &   SONS,    NEW  YORK. 
CHAPMAN  &  HALL,  Limited,  LONDON. 


MATHEMATICAL    MONOGRAPHS. 

EDITED    BV 

MANSFIELD   MERRIMAN  and  ROBERT   S.   WOODWARD. 


No.  10. 


THE    SOLUTION    OF 
EQUATIONS. 


BY 

MANSFIELD    MERRIMAN, 

Professor  of  Civil  Engineering  in  Lehigh  University. 


FOURTH    EDITION.   ENLARGED. 
FIRST    THOUSAND. 


NEW  YORK: 

JOHN   WILEY    &    SONS. 

London-    CHAPMAN  &   HALL,    Limited. 

1906. 


Copyright,  1896, 

BY 

MANSFIELD    MERRIMAN  and  ROBERT   S.   WOODWARD 

UNDER   THE    T.TLE 

HIGHER    MATHEMATICS. 

First  Edition,  September,  1896. 
Second  Edition,  January,  1898. 
Third  Edition,  August,  1900. 
Fourth  Edition,   January,  1906. 


'     «  «  , 

.*  c    t 


ROBRRT  DRUMMONP,    TWINTPF,    VFW   VORK. 


EDITORS'   PREFACE. 


The  volume  called  Higher  Mathematics,  the  first  edition 
of  which  was  published  in  1896,  contained  eleven  chapters  by 
eleven  authors,  each  chapter  being  independent  of  the  others, 
but  all  supposing  the  reader  to  have  at  least  a  mathematical 
training  equivalent  to  that  given  in  classical  and  engineering 
colleges.  The  publication  of  that  volume  is  now  discontinued 
and  the  chapters  are  issued  in  separate  form.  In  these  reissues 
it  will  generally  be  found  that  the  monographs  are  enlarged 
by  additional  articles  or  appendices  which  either  amplify  the 
former  presentation  or  record  recent  advances.  This  plan  of 
publication  has  been  arranged  in  order  to  meet  the  demand  of 
teachers  and  the  convenience  of  classes,  but  it  is  also  thought 
that  it  may  prove  advantageous  to  readers  in  special  lines  of 
mathematical  literature. 

It  is  the  intention  of  the  publishers  and  editors  to  add  other 
monographs  to  the  series  from  time  to  time,  if  the  call  for  the 
same  seems  to  warrant  it.  Among  the  topics  which  are  under 
consideration  are  those  of  elUptic  functions,  the  theory  of  num- 
bers, the  group  theory,  the  calculus  of  variations,  and  non- 
EucUdean  geometry;  possibly  also  monographs  on  branches  of 
astronomy,  mechanics,  and  mathematical  physics  may  be  included. 
It  is  the  hope  of  the  editors  that  this  form  of  pubHcation  may 
tend  to  promote  mathematical  study  and  research  over  a  wider 
field  than  that  which  the  former  volume  has  occupied. 

December,  1905. 


Ill 


236371 


AUTHOR'S   PREFACE. 


The  following  pages  are  designed  as  supplementary  to  the 
discussions  of  equations  in  college  text-books,  and  several  methods 
of  solution  not  commonly  given  in  such  works  are  presented 
and  exemplified.  The  aim  kept  in  view  has  been  that  of  the 
determination  of  the  numerical  values  of  the  roots  of  numerical 
equations,  and  algebraic  analysis  has  been  used  only  to  further 
this  end.  Historical  references  are  given,  problems  stated  as 
exercises  for  the  student,  and  the  attempt  has  everywhere  been 
made  to  present  the  subject  clearly  and  concisely.  The  volume 
has  not  been  written  for  those  thoroughly  conversant  with  the 
theory  of  equations,  but  rather  for  students  of  mathematics, 
computers,  and  engineers. 

This  edition  has  been  enlarged  by  the  addition  of  five  articles 
which  render  the  former  treatment  more  complete  and  also  give 
recent  investigations  regarding  the  expression  of  roots  in  series. 
While  not  designed  for  college  classes,  it  is  hoped  that  the  book 
may  prove  useful  to  postgraduate  students  in  mathematics, 
physics  and  engineering,  and  also  tend  to  promote  general  interest 
in  mathematical  science. 

South  Bethlehem,  Pa., 
December,  1905. 


CONTENTS. 


Art.  I.  Introduction Page  i 

2.  Graphic  Solutions 3 

3.  The  Regula  Falsi 5 

4.  Newton's  Approximation  Rule 6 

5.  Separation  of  the  Roots 8 

6.  Nutvlerical  Algebraic  Equations 10 

7.  Transcendental  Equations 13 

8.  Algebraic  Solutions 15 

9.  The  Cubic  Equation 17 

10.  The  Quartic  Equation 19 

11.  QuEsiTic  Equations 21 

12.  Trigonometric  Solutions 24 

13.  Real  Roots  by  Series 27 

14.  Computation  of  All  Roots       .... 28 

15.  Roots  of  Unity 31 

16.  Solutions  by  Maclaurin's  Series 33 

17.  Symmetric  Functions  of  Roots 37 

18.  Logarithmic  Solutions 39 

19.  Infinite  Equations 43 

20.  Notes  and  Problems 45 

Index 47 


THE    SOLUTION    OF    EQUATIONS. 


Art.  1.    Introduction. 

The  science  of  algebra  arose  in  the  efforts  to  solve  equations. 
Indeed  algebra  may  be  called  the  science  of  the  equation,  since 
the  discussion  of  equalities  and  the  transformation  of  forms  into 
simpler  equivalent  ones  have  been  its  main  objects.  The  solu- 
tion of  an  equation  containing  one  unknown  quantity  consists 
in  the  determination  of  its  value  or  values,  these  being  called 
roots.  An  algebraic  equation  of  degree  n  has  n  roots,  while  tran- 
scendental equations  often  have  an  infinite  number  of  roots.  The 
object  of  the  following  pages  is  to  present  and  exemplify  convenient 
methods  for  the  determination  of  the  numerical  values  of  the 
roots  of  both  kinds  of  equations,  the  real  roots  receiving  special 
attention  because  these  are  mainly  required  in  the  solution  of 
problems  in  physical  science. 

An  algebraic  equation  is  one  that  involves  only  the  opera- 
tions of  arithmetic.  It  is  to  be  first  freed  from  radicals  so  as 
to  make  the  exponents  of  the  unknown  quantity  all  integers; 
the  degree  of  the  equation  is  then  indicated  by  the  highest  ex- 
ponent of  the  unknown  quantity.  The  algebraic  solution  of  an 
algebraic  equation  is  the  expression  of  its  roots  in  terms  of 
the  literal  coefficients ;  this  is  possible,  in  general,  only  for  linear, 
quadratic,  cubic,  and  quartic  equations,  that  is,  for  equations 
of  the  first,  second,  third,  and  fourth  degrees.  A  numerical 
equation  is  an  algebraic  equation  having  all  its  coefficients  real 
numbers,  either  positive  or  negative.     For  the   four  degrees 


2  THE    SOLUTION    OF    EQUATIONS. 

above  mentioned  the  roots  of  numerical  equations  may  be 
computed  from  the  formulas  for  the  algebraic  solutions,  unless 
they  fall  under  the  so-called  irreducible  case  wherein  real 
quantities  are  expressed  in  imaginary  forms. 

An  algebraic  equation  of  the  n^^  degree  may   be  written 
with  all  its  terms  transposed  to  the  first  member,  thus : 

X"  +  a,x"-'  +  a,x'"-'  +  .  .  .  +  a„_,x  +  a„  =  o, 

and,  for  brevity,  the  first  member  will  be  called  /{x)  and  the 
equation  be  referred  to  as/{x)  =  o.  The  roots  of  this  equa- 
tion are  the  values  of  x  which  satisfy  it,  that  is,  those  values  of 
X  that  reduce /(x)  to  o.  When  all  the  coefficients  a^,  a^, .  .  .a„ 
are  real,  as  will  always  be  supposed  to  be  the  case,  Sturm's 
theorem  gives  the  number  of  real  roots,  provided  they  are  un- 
equal, as  also  the  number  of  real  roots  lying  between  two 
assumed  values  of  x,  while  Horner's  method  furnishes  a  con- 
venient process  for  obtaining  the  values  of  the  roots  to  any 
required  degree  of  precision. 

A  transcendental  equation  is  one  involving  the  operations 
of  trigonometry  or  of  logarithms,  as,  for  example,  x  -\-  cos;i;  =  o, 
or  a^"" -\- xir' =  O.  No  general  method  for  the  literal  solution 
of  these  equations  exists  ;  but  when  all  known  quantities  are 
expressed  as  real  numbers,  the  real  roots  may  be  located  and 
computed  by  tentative  methods.  Here  also  the  equation  may 
be  designated  as/(;f)  =  o,  and  the  discussions  in  Arts.  2-5  will 
apply  equally  well  to  both  algebraic  and  transcendental  forms. 
The  methods  to  be  given  are  thus,  in  a  sense,  more  valuable 
than  Sturm's  theorem  and  Horner's  process,  although  for 
algebraic  equations  they  may  be  somewhat  longer.  It  should 
be  remembered,  however,  that  algebraic  equations  higher  than 
the  fourth  degree  do  not  often  occur  in  physical  problems,  and 
that  the  value  of  a  method  of  solution  is  to  be  measured  not 
merely  by  the  rapidity  of  computation,  but  also  by  the  ease 
with  which  it  can  be  kept  in  mind  and  applied. 

Prob.  I.  Reduce  the  equation  (a  +  x)^  -{-  (a  —  x)^  =  2b  to  an 
equation  having  the  exponents  of  the  unknown  quantity  all  integers. 


GRAPHIC    SOLUTIONS, 


'6 


c'',     from    which    the    two 


Art.  2.    Graphic  Solutions. 

Approximate  values  of  the  real  roots  of  two  simultaneous 
algebraic  equations  may  be  found  by  the  methods  of  plane 
analytic  geometry  when  the  coefficients  are  numerically 
expressed.     For  example,  let  the  given  equations  be 

x"  -{-y"  =  a\  x'  —  bx  =y  —  cy, 

the  first  representing  a  circle  and  the  second  a  hyperbola. 
Drawing  two  rectangular  axes  OX  and  OY,  the  circle  is  de- 
scribed from  O  with  the  radius  a.  The  coordinates  of  the 
center  of  the  hyperbola  are  found  to  be  OA  =  \b  and  AC  ^\c, 

while    its    diameter    BD  =  ^ b'  — 

branches    may  be   described. 

The  intersections  of  the  circle 

with  the   hyperbola  give   the 

real    values    of   x   and  y.  %  If 

a=i  I,  b  ^  ^,  and  ^  =  3,  there 

are  but  two  real  values  for  x 

and    two    real    values    for   j/, 

since  the  circle  intersects  but 

one  branch  of  the  hyperbola  ; 

here    Om    is  the  positive  and 

Op  the  negative  value  of  x,  while  7nn  is  the  positive  and  pq 

the  negative  value  oi  y.     When   the  radius  a  is  so  large  that 

the  circle  intersects  both  branches  of  the  hyperbola  there  are 

four  real  values  of  both  x  and  y. 

By  a  similar  method  approximate  values  of  the  real  roots  of 
an  algebraic  equation  containing  but  one  unknown  quantity  may 
be  graphically  found.  For  instance,  let  the  cubic  equation 
x^  -^  ax  —  b  ^  o  be  required  to  be  solved.*  This  may  be 
written  as  the  two  simultaneous  equations 

y  ^=z  x"",  y  =  —  ax  -\-  b, 

*  See  Proceedings  of  the  Engineers'  Club  of  Philadelphia,   1884,  V'.l.  IV, 
pp.  47-49 


THE    SOLUTION    OF    EQUATIONS. 


and  the  graph  of  each  being  plotted,  the  abscissas  of  their 
points  of  intersection  give  the  real  roots  of  the  cubic.     The 

curve  J/  =  x^  should  be  plotted  upon 
cross-section  paper  by  the  help  of  a 
table  of  cubes ;  then  OB  is  laid  off 
equal  to  d,  and  OC  equal  to  a/d,  tak- 
ing care  to  observe  the  signs  of  a  and 
d.  The  line  joining  B  and  C  cuts 
the  curve  at  p,  and  hence  g/>  is  the 
real  root  oi  x^  -\-  ax  —  b  =  o.  If  the 
cubic  equation  have  three  real  roots  the  straight  line  BC  will 
intersect  the  curve  in  three  points. 

Some  algebraic  equations  of  higher  degrees  may  be  graphic- 
ally solved  in  a  similar  manner.  For  the  quartic  equation 
s" -\- Az' -\- Bz  —  C  ^  o,  \\.  is  best  to  put  s  =  A^x,  and  thus 
reduce  it  to  the  form  x* -{- x'' -\- dx  —  c  =  o ;  then  the  two 
equations  to  be  plotted  are 

J/  ^  X*  -{-  x'',        y  ■=  —  bx  -\-  c, 

the  first  of  which  may  be  drawn  once  for  all  upon  cross-section 
paper,  while  the  straight  line  represented  by  the  second  may 
be  drawn  for  each  particular  case,  as  described  above.* 

This  method  is  also  applicable  to  many  transcendental  equa- 
tions ;  thus  for  the  equation  .^;f  —  ^sin;f  =  o  it  is  best  to 
write  ax  —  sin;r  =  o ;  then  y  :=  smx  is  readily  plotted  by  help 
of  a  table  of  sines,  while  y  ^=  ax  is  a  straight  line  passing 
through  the  origin.  In  the  same  way  a"^  —  x'^  ^^  o  gives  the 
curve  represented  byj:=rt'*and  the  parabola  represented  by 
y  =  x"^,  the  intersections  of  which  determine  the  real  roots  of 
the  given  equation. 

Prob.  2.  Devise  a  graphic  solution  for  finding  approximate 
values  of  the  real  roots  of  the  equation  x''-\-  ax^-\-  bx''-{-  ex  +  d  =o. 

Prob.  3.  Determine  graphically  the  number  and  the  approximate 
values  of  the  real  roots  of  the  equation  arc  x  —  2>  sin  x  =  o. 
(Ans. — Six  real  roots,  x  =  ±  159°,  ±  430°,  and  ±  456°.) 

*  For  an  extension  of  this  method  to  the  determination  of  imaginary  roots, 
see  Phillips  and  Beebe's  Graphic  Algebra,  New  York,  1S82. 


THE    REGULA    FALSI. 


OA 


Art.  3.     The  Regula  Falsi. 

One  of  the  oldest  methods  for  computing  the  real  root  of 
an  equation  is  the  rule  known  as  "  regula  falsi,"  often  called 
the  method  of  double  position.*  It  depends  upon  the  princi- 
ple that  if  two  numbers  x^  and  x^  be  substituted  in  the  expres- 
sion y"(;r),  and  if  one  of  these  renders  f{x)  positive  and  the  other 
renders  it  negative,  then  at  least  one  real  root  of  the  equation 
f[x)  =  o  lies  between  x^  and  x^.  Let  the  figure  represent  a 
part  of  the  real  graph  of  the  equation  y  =y(,r).  The  point  X, 
where  the  curve  crosses  the  axis  of  abscissas,  gives  a  real  root 
OX  of  the  equation  f{x)  =  o.  Let  OA  and  OB  be  inferior  and 
superior  limits  of  the  root  OX  which  are  determined  either  by 
trial  or  by  the  method  of  Art,  5. 
Let  Aa  and  Bb  be  the  values  of 
/{x)  corresponding  to  these  limits. 
Join  ad,  then  the  intersection  C  of 
the  straight  line  al?  with  the  axis 
OB  gives  an  approximate  value 
OC  for  the  root.  Now  compute 
Cc  and  join  ac,  then  the  intersection  D  gives  a  value  OD  which 
is  closer  still  to  the  root  OX. 

Let  ,r,  and  x^  be  the  assumed  values  OA  and  OB,  and  let 
/{x^)  and/(-rj)  be  the  corresponding  values  o(  /(x)  represented 
by  Aa  and  Bd,  these  values  being  with  contrary  signs.  Then 
from  the  similar  triangle  AaC  and  BdC  the  abscissa  OC  is 

'''~     A^.)-A^\)       ■*'""^At'J-/(^,)  ""'^"^  At-.)  -/K)  • 

By  a  second  application  of  the  rule  to  x^  and  x^,  another  value 

x^  is  computed,  and  by  continuing  the  process  the  value  of  x 

can  b^  obtained  to  any  required  degree  of  precision. 

As  an  example  \e\.  f{x)  =1  x^  -\-  ^x''  -|-  7  =  o.     Here  it  may 

be  found  by  trial  that  a  real  root  lies  between  —2  and  —  1.8. 

*  This  originated  in  India,  and  its  first  publication  in  Europe  was  by  Abra- 
ham ben  Esra,  in  1130.  See  Matthiesen,  Grundziige  der  antiken  und  moder- 
nen  Algebra  der  litteralen  Gieichungen,  Leipzig,  1878. 


6  THE    SOLUTION    OF    EQUATIONS. 

For  ^,  =  —  2,f{x^  =  —  5,  and  for  x^  =  —i.S,/{x,)  =  +4.304; 
then  by  the  regula  falsi  there  is  found  x^  =  —  1.90  nearly. 
Again,  for  x^  =  —  1.90,  /{x^)  =  -{-  0.290,  and  these  combined 
with  x^  and  /{x^)  give  x^  =  —  1.906,  which  is  correct  to  the 
third  decimal. 

As  a  second  example  let  /{x)  =  arc;r  —  sin;ir  —  0.5  =  o. 
Here  a  graphic  solution  shows  that  there  is  but  one  real  root, 
and  that  the  value  of  it  lies  between  85°  and  86°.  For  x^=  85°, 
/(•^i)  =  —  0.01266,  and  for  x^  =  86°, /(,r,)  —  -f  0.00342  ;  then 
by  the  .rule  ;ir3  =  85"  44',  which  gives /(;tr3)  =  —  0.00090.  Again, 
combining  the  values  for  x^  and  x^  there  is  found  x^  =  85°  47'. 
which  gives  /{x^)  =  —  0.00009.  Lastly,  combining  the  values 
for  x^  and  x^  there  is  found  ;irj,  =  85 "  47'.4,  /vhich  is  as  close  an 
approximation  as  can  be  made  with  five-place  tables. 

In  the  application  of  this  method  it  is  to  be  observed  that 
the  signs  of  the  values  of  x  and  /(x)  are  to  be  carefully  re- 
garded, and  also  that  the  values  of  /(x)  to  be  combined  in  one 
operation  should  have  opposite  signs.  For  the  quickest 
approximation  the  values  of /(:r)  to  be  selected  should  be  those 
having  the  smallest  numerical  values. 

Prob.  4.  Compute  by  the  regula  falsi  the  real  roots  of  Jf^  — 0.25=0. 
Also  those  of  jc^  +  sin  2Jc  =  o. 

Art.  4.    Newton's  Approximation  Rule. 

Another  useful  method  for  approximating  to  the  value  of 
the  real  root  of  an  equation  is  that  devised  by  Newton  in  1666.'^ 

^ct---'  If  J/  =/{x)  be  the  equation  of  a 
curve,  OX  in  the  figure  represents  a 
real  root  of  the  equation  /(x)  =  o. 
Let  OA  be  an  approximate  value  of 
OX,  and  Aa  the  corresponding  value 
of/(;ir).  At  a  let  aB  be  drawn  tangent 
to  the  curve ;  then  OB  is  another  approximate  value  of  OX.   0 

*  See  Analysis  per  equationes  numero  terminorum  infinitas,  p.  26g,  Vol.  I 
of  Horsely's  edition  of  Newton's  works  (London,  1779),  where  the  method  is 
given  in  a  somewhat  different  form. 


NEWTON  S    APPROXIMATION    RULE.  ( 

Let  Bb  be  the  value  of  f{x)  corresponding  to  OB,  and  at  b 
let  the  tangent  bC  be  drawn ;  then  OC  is  a  closer  approxima- 
tion to  OX,  and  thus  the  process  may  be  continued. 

Let/'(,i-)  be  the  first  derivative  oi  f{x)\  or,f'{x)  —  df{x)/dx. 
For  X  ^^  x^  —  OA  in  the  figure,  the  value  of  /{x^)  is  the  ordi- 
nate Aa,  and  the  value  of  /'(^,)  is  the  tangent  of  the  angle 
aBA  ;  this  tangent  is  also  Aa/AB.  Hence  AB  —  f\x^/f\x^, 
and  accordingly  OB  and  OC  are  found  by 

;,_;,_    Ml  X      -X       -      -^^"l^^- 

which  is  Newton's  approximation  rule.  By  a  third  application 
to  x^  the  closer  value  x^  is  found,  and  the  process  may  be  con- 
tinued to  any  degree  of  precision  required. 

For  example,  let  f{x)  =  x^  -{-  ^x'^  -|-  7  =  O.  The  first  deriv- 
ative isy"'(^)  =  ^x*  -f-  \ox.  Here  it  may  be  found  by  trial  that 
—  2  is  an  approximate  value  of  the  real  root.  For  x^^=  —  2 
f{^\)  =^  —  5'  "^'""^ /X-*"!)  =  60,  whence  by  the  rule  x^  =  —  1.92. 
Now  for  x^  =  —  1.92  are  found  f{x^  =  —  0.6599  and 
/'(x^)  =  29  052,  whence  by  the  rule  x^  =  —  I.906,  which  is 
correct  to  the  third  decimal. 

As  a  second  example  let  /{x)  =  x^  -{-  4sin;ir  =  o.  Here 
the  first  derivative  is /'(,r)  =  2,i' -f-4cos;f.  An  approximate 
value  of  X  found  either  by  trial  or  by  a  graphic  solution  is 
;r=  — 1.94,  corresponding  to  about  —  1 1 1°09'.  For  ;ir,  =  — 1.94, 
/(x^)  =  0.03304  and  /\xj  =  —  5.323,  whence  by  the  rule 
x^=  —  1.934.  By  a  second  application  x^  =  —  1.9328,  which 
corresponds  to  an  angle  of  —  110°  54i-'. 

In  the  application  of  Newton's  rule  it  is  best  that  the 
assumed  value  of  ,r,  should  be  such  as  to  render  y'(;r,)  as  small 
as  possible,  and  also/'(,r,)  as  large  as  possible.  The  method 
will  fail  if  the  curve  has  a  maximum  or  minimum  between  a 
and  b.  It  is  seen  that  Newton's  rule,  like  the  regula  falsi, 
applies  equally  well  to  both  transcendental  and  algebraic  equa- 
tions, and  moreover  that  the  rule  itself  is  readily  kept  in  mind 
by  help  of  the  diagram. 


8  THE    SOLUTION    OF    EQUATIONS. 

Prob.  5.  Compute  by  Newton's  rule  the  real  roots  of  the  alge- 
braic equation  x*  —  "jx  -\-  6  =  o.  Also  the  real  roots  of  the  trans- 
cendental equation  sin  x  -\-  arc  :*:  —  2  =  o. 


Art.  5.    Separation  of  the  Roots. 

The  roots  of  an  equation  are  of  two  kinds,  real  roots  and 
imaginary  roots.  Equal  real  roots  may  be  regarded  as  a  spe- 
cial class,  which  he  at  the  limit  between  the  real  and  the  imagi- 
nary. If  an  equation  has p  equal  roots  of  one  value  and  g  equal 
roots  of  another  value,  then  its  first  derivative  equation  has 
p  —  I  roots  of  the  first  value  and  g  —i  roots  of  the  second 
value,  and  thus  all  the  equal  roots  are  contained  in  a  factor 
common  to  both  primitive  and  derivative.  Equal  roots  may 
hence  always  be  readily  detected  and  removed  from  the  given 
equation.  For  instance,  let  x*  —  gx"  -\-  4.x  -\-  12  =  o,  of  which 
the  derivative  equation  is  4Jir'  —  i8;r-|-4  =  o;  as;ir  —  2  is  a 
factor  of  these  two  equations,  two  of  the  roots  of  the  primitive 
equation  are  -f-  2. 

—  The  problem  of  determining  the  number  of  the  real  and 
imaginary  roots  of  an  algebraic  equation  is  completely  solved 
by  Sturm's  theorem.  If,  then,  two  values  be  assigned  to  x  the 
number  of  real  roots  between  those  limits  is  found  by  the  same 
theorem,  and  thus  by  a  sufficient  number  of  assumptions  limits 
may  be  found  for  each  real  root.  As  Sturm's  theorem  is  known 
to  all  who  read  these  pages,  no  applications  of  it  will  be  here 
given,  but  instead  an  older  method  due  to  Hudde  will  be 
presented  which  has  the  merit  of  giving  a  comprehensive  view 
of  the  subject,  and  which  moreover  applies  to  transcendental 
as  well  as  to  algebraic  equations.* 

If  any  equation  y -=1  f{x)  be  plotted  with  values  of  x  as 
abscissas  and  values  oi  y  as  ordinates,  a  real  graph  is  obtained 
whose  intersections  with  the  axis  (^JTgive  the  real  roots  of  the 

*  Devised  by  Hudde  in  1659  and  published  by  Rolie  in  1690.     See  CEuvres 
de  Lagrange,  Vol.  VIII,  p.  190. 


SEPARATION    OF    THE    ROOTS. 


equa/ion y"(;tr)  =  o.  Thus  in  the  figure  the  three  points  marlced 
X  gi>  'e  three  values  OX  for  three  real  roots.  The  curve  which 
repr'  sents  j  ^  f{x)  has  points  of  maxima  and  minima  marked 
A,  3. id  inflection  points  marked  B.     Now  let  the  first  deriva- 


tive equation  dy/dx=^  f\x)  be  formed  and  be  plotted  in  the 
same  manner  on  the  axis  O' X .  The  condition /'(-r)=  o  gives 
the  abscissas  of  the  points  A^  and  thus  the  real  roots  OX'  give 
limits  separating  the  real  roots  oi  f{pc)  ==  o.  To  ascertain  if  a 
real  root  OX  lies  between  two  values  of  O' X'  these  two  values 
are  to  be  substituted  m  fix):  if  the  signs  oi  f{x)  are  unlike  in 
the  two  cases,  a  real  root  of  fix)  =  o  lies  between  the  two 
limits;  if  the  signs  are  the  same,  a  real  root  does  not  lie  between 
those  limits. 

In  like  manner  if  the  second  derivative  equation,  that  is, 
dy/dx''  =  /"{x),  be  plotted  on  O'X",  the  intersections  give 
limits  which  separate  the  real  roots  of /"'(-^)=o.  It  is  also 
seen  that  the  roots  of  the  second  derivative  equation  are  the 
abscissas  of  the  points  of  inflection  of  the  curve/  =  /(x). 

To  illustrate  this  method  let  the  given  equation  be  the 
quintic  /{x)  =  x^  —  5^'  -{-6x  -\-  2  =  o.  The  first  derivativ^e 
equation  is  f\x)  =  $x*  —i$x^'}-6  =  o,  the  roots  of  which  are 
•approximately  —  1.59,  —0.69,  +0.69.  -\-  1.59.  Now  let  each 
of  these  values  be  substituted  for  x  in  the  given  quintic,  as  also 
the  values  —  00  ,  o,  and  -\-  00  ,  and  let  the  corresponding  values 
of /(;r)  be  determined  as  follows  : 


10  THE    SOLUTION    OF    EQUATIONS. 

;r  =  — 00,     —1.59,     —0.69,     o,     +0.69,     +1.59,     +00; 
f{x)=-^,     +2.4,       -0.6,  +2,     +47.       +1-6,       +CO. 

Since  f{x)  changes  sign  between  x^=^ —  00  and  x^  =  —  i-59> 
one  real  root  lies  between  these  limits  ;  smce  f{x)  changes  sign 
between  ;r,  =  —  i .59  and  x^^=  —  0.69,  one  real  root  lies  between 
these  limits  ;  since  y"(;tr)  changes  sign  between  x^^  —  0.69  and 
x^  =  O,  one  real  root  lies  between  these  limits;  since /(;ir)  does 
not  change  sign  between  ^1:3  =  0  and  ;tr^  =  00  ,  a  pair  of  imagi- 
nary roots  is  indicated,  the  sum  of  which  lies  between  -\-  0.69 
and  00 . 

As  a  second  example  let  f{x)  =^  e'  —  ^—4  =  0.  The  first 
derivative  equation  is  f'{x)  =:  r*  —  2^*  =  O,  which  has  two 
roots  ^  =  |-  and  ,?^  =  o,  the  latter  corresponding  to  ;ir  =  —  co . 
For  X  :=  —  <^  ,  f{x)  is  negative;  for  e^  =.^,  f{x)  is  negative  ;  for 
X  ^  -\-  CO  ,  f{x)  is  negative.  The  equation  e^  —  ^  —  4  =  0 
has,  therefore,  no  real  roots. 

When  the  first  derivative  equation  is  not  easily  solved,  the 
second,  third,  and  following  derivatives  may  be  taken  until  an 
equation  is  found  whose  roots  may  be  obtained.  Then,  by 
working  backward,  limits  may  be  found  in  succession  for  the 
roots  of  the  derivative  equations  until  finally  those  of  the 
primative  are  ascertained.  In  many  cases,  it  is  true,  this  proc- 
ess may  prove  lengthy  and  difficult,  and  in  some  it  may  fail 
entirely;  nevertheless  the  method  is  one  of  great  theoretical 
and  practical  value. 

Prob.  6.  Show  that  e"  +  e'^''  —4  =  0  has  two  real  roots,  one 
positive  and  one  negative. 

Prob.  7.  Show  that  x^  -\-  x -\-  \  ^=-  o  has  no  real  roots;  also  that 
x^  —  X  —  \  =0  has  two  real  roots,  one  positive  and  one  negative. 

Art.  6.     Numerical  Algebraic  Equations. 

An  algebraic  equation  of  the  ;?"'  degree  may  be  written 
with  all  its  terms  transposed  to  the  first  member,  thus: 

;r"  +  a^x""-^  +  a^x''-"-  +  .  .  .  -f  a„_^x  -|-  «„  =  O ; 


NUMERICAL    ALGEBRAIC    EQUATIONS.  11 

and  if  all  the  coefficients  and  the  absolute  term  are  real  num- 
bers, this  is  commonly  called  a  numerical  equation.  The  first 
member  may  for  brevity  be  denoted  by/(^,r)  and  the  equation 
itself  by/(^-)  =  o. 

The  following  principles  of  the  theory  of  algebraic  equations 
with  real  coefficients,  deduced  in  text-books  on  algebra,  are 
here  recapitulated  for  convenience  of  reference  : 

(i)  If  .r,  is  a  root  of  the  equation, /(x)  is  divisible  by  x  —  x^; 
and  conversely,  ii /{x)  is  divisible  by  x  —x^,  then  x\  is  a  root  of  the 
equation. 

(2)  An  equation  of  the  «*''  degree  has  n  roots  and  no  more. 

(3)  If  jc, ,  ^2,  .  .  .  x,i  are  the  roots  of  the  equation,  then  the  prod- 
uct (x  —  x^){x  —  xj  .  .  .  (x  —  x,i)  is  equal  to/(a~). 

(4)  The  sum  of  the  roots  is  equal  to  —  «,;  the  sum  of  the  prod- 
ucts of  the  roots,  taken  two  in  a  set,  is  equal  to  -{-  a^;  the  sum  of 
the  products  of  the  roots,  taken  three  in  a  set,  is  equal  to  —  a^;  and 
so  on.  The  product  of  all  the  roots  is  equal  to  —  a,,  when  fi  is 
odd,  and  to  +  a„  when  u  is  even. 

(5)  The  equation /(a:)  =  o  may  be  reduced  to  an  equation  lack- 
ing its  second  term  by  substituting _>'  —  a^/fl  for  x.* 

(6)  If  an  equation   has   imaginary  roots,  they  occur  in  pairs   of 

the  form  /  ±  ^/  where  i  represents  y  —  i. 

(7)  An  equation  of  odd  degree  has  at  least  one  real  root  whose 
sign  is  opposite  to  that  of  a,i- 

(8)  An  equation  of  even  degree,  having  a„  negative,  has  at  least 
two  real  roots,  one  being  positive  and  the  other  negative. 

(9)  A  complete  equation  cannot  have  more  positive  roots  than 
variations  in  the  signs  of   its  terms,  nor  more  negative   roots  than      ^ 
permanences  in  signs.     If  all  roots  be  real,  there  are  as  many  posi- 
tive roots  as  variations,  and  as  many  negative  roots  as  permanences. f 

(10)  In  an  incomplete  equation,  if  an  even  number  of  terms, 
say  2m,  are  lacking  between  two  other  terms,  then  it  has  at  least  2m 

*  By  subsiitutingy  +/V  +  ?  for  x.  the  quantities/ and  ^  maybe  determined 
so  as  to  remove  the  second  and  third  terms  by  means  of  a  quadratic  equation, 
the  second  and  fourth  terms  by  means  of  a  cubic  equation,  or  the  second  and 
fifth  terms  by  means  of  a  quartic  equation. 

+  The  'aw  deduced  by  Harriot  in  1631  and  by  Descartes  in  1639. 


y 


12  THE    SOLUTION    OF    EQUATIONS, 

imaginary  roots;  if  an  odd  number  of  terms,  say  2m  +  i,  are  lacking 
/      between  two   other  terms,  then  it  has  at  least  either   zni  +  2  or  zm 
imaginary  roots,  according   as   the   two   terms   have   like  or  unlike 
signs.* 

(11)  Sturm's  theorem  gives  the  number  of  real  roots,  provided 
/    that  they  are  unequal,  as  also  the  number  of  real  roots  lying   be- 
tween two  assumed  values  of  x. 

(12)  If  ar  is  the  greatest  negative  coefficient,  and  if  a^  is  the 
greatest  negative  coefficient  after  .t  is  changed  into  —  .v,  then  all 
real  roots  lie  between  the  limits  ar  -\-  i  and  —  {as  +  i). 

(13)  If  ah  is  the  first  negative  and  a^  the  greatest   negative  co- 
I 

efficient,  then  ar'  -)-  i  is  a  superior  limit  of  the  positive  roots.     If 

ak  be  the  first  negative  and  as  the  greatest  negative  coefficient  after 

I 

jc  is  changed  into  —  x,  then  as  +  '  is  a  numerically  superior  limit 
of  the  negative  roots. 

(14)  Inferior  limits  of  the  positive  and  negative  roots  may  be 
found  by  placing  x  =  s~'  and  thus  obtaining  an  equation  /{z)  =  o 
whose  roots  are  the  reciprocals  of  /{x)  =  o. 

(15)  Horner's  method,  using  the  substitution  x  —  z  —  r  where  r 
is  an  approximate  value  of  x\  ,  enables  the  real  root  x^  to  be  com- 
puted to  any  required  degree  of  precision. 

The  application  of  these  principles  and  methods  will  be 
familiar  to  all  who  read  these  pages.  Horner's  method  may 
be  also  modified  so  as  to  apply  to  the  computation  of  imagi- 
nary roots  after  their  approximate  values  have  been  found. f 
The  older  method  of  Hudde  and  Rolle,  set  forth  in  Art.  5,  is 
however  one  of  frequent  convenient  application,  for  such  alge- 
braic equations  as  actually  arise  in  practice.  By  its  use, 
together  with  principles  (13)  and  (14)  above,  and  the  regula 
falsi  of  Art.  3,  the  real  roots  may  be  computed  without  any 
.assumptions  whatever  regarding  their  values. 

For  example,  let  a  sphere  of  diameter  D  and  specific  gravity 

*  Established  by  DuGua;  see  Memoirs  Paris  Academy,  1741,  pp.  435-494. 

f  Sheffler,  Die  Auflosung  der  algebraischen  und  transzendenten  Gleichung- 
-  en,  Braunschweig,  1859;  and  Jelink,  Die  Auflosung  der  hoheren  numerischen 
wQleichungen,  Leipzig,  1865. 


TRANSCENDENTAL  EQUATIONS.  13 

g  float  in  water,  and  let  it  be  required  to  find  the  depth  of  im- 
mersion. The  solution  of  the  problem  gives  for  the  depth  x 
the  cubic  equation 

x'  -  ^Dx'  +  Wg  =  o. 

•  As  a  particular  case  let  Z)  =  2  feet  and  ^=0.65;    then  the 

equation 

x^  —  ix''  -|-  2.6  =  o 

is  to  be  solved.  The  first  derivative  equation  is  t^x^  —  6x  =  o 
whose  roots  are  o  and  2.  Substituting  these,  there  is  found 
one  negative  root,  one  positive  root  less  than  2,  and  one  posi- 
tive root  greater  than  2.  The  physical  aspect  of  the  question 
excludes  the  first  and  last  root,  and  the  second  is  to  be  computed. 
By  (13)  and  (14)  an  inferior  limit  of  this  root  is  about  0.5,  so 
that  it  lies  between  0.5  and  2.  For  x^  =  0.5, /(Xj)  =  -4"  i-975r 
and  for  x,  —  2,/(;r,)  =  —1.4;  then  by  the  regula  falsi  -^3=1.35. 
For  x^  =i-35,/(-*'s)  =  —  0,408,  and  combining  this  with  x,  the 
regula  falsi  gives  x^  =  1.204  feet,  which,  except  in  the  last 
decimal,  is  the  correct  depth  of  immersion  of  the  sphere. 

Prob.  8.  The  diameter  of  a  water-pipe  whose  length  is  200  feet 
and  which  is  to  discharge  100  cubic  feet  per  second  under  a  head 
of  10  feet  is  given  by  the  real  root  of  the  quintic  equation. 
x^  —  38JC  —  101  =  o.     Find  the  value  of  x. 

Art.  7.    Transcendental  Equations. 

Rules  (i)  to  (15)  of  the  last  article  have  no  application  to 
trigonometrical  or  exponential  equations,  but  the  general  prin- 
ciples and  methods  of  Arts.  2-5  may  be  always  used  in 
attempting  their  solution.  Transcendental  equations  may 
have  one,  many,  or  no  real  roots,  but  those  arising  from  prob- 
lems in  physical  science  must  have  at  least  one  real  root.  Two 
examples  of  such  equations  will  be  presented. 

A  cylinder  of  specific  gravity^  floats  in  water,  and  it  is 
required  to  find  the  immersed  arc  of  the  circumference.  If 
this  be  expressed  in  circular  measure  it  is  given  by  the  trans- 
cedental  equation 

/{x)  ^  X  —  sin  ;r  —  2;r^  =  O. 


14  THE    SOLUTION    OF    EQUATIONS, 

The  first  derivative  equation  is  i  — cos  ;ir  =  o,  whose  root  is 
any  even  multiple  of  27t.  Substituting  such  multiples  in  y"(^) 
it  is  found  that  the  equation  has  but  one  real  root,  and  that 
this  lies  between  o  and  2n\  substituting  \7t,  ^tt,  and  rr  for  x,  it 
is  further  found  that  this  root  lies  between  f;r  and  tt. 

As  a  particular  case  let  ^  — .  0.424,  and  for  convenience  in 
using  the  tables  let  x  be  expressed  in  degrees;  then 

/{x)  —  X  —  57°  .2958  sin  X  ~  152°  .64. 

Now  proceeding  by  the  regula  falsi  (Art.  3)  let  x^  =  180°  and 
-^.=  135°,  giving /(;ir,)  =+ 27°  .36  and/(j;J  =-58°  .16,  whence 
jTj  =  166°.  For  x^-=.  166°,  f{x^  =  — o"  .469,  and  hence  166°  is  an 
approximate  value  of  the  root.  Continuing  the  process,  x  is 
found  to  be  i66°.237,  or  in  circular  measure  ;r=2, 9014  radians. 

As  a  second  example  let  it  be  required  to  find  the  horizon- 
tal tension  of  a  catenary  cable  whose  length  is  22  feet,  span  20 
feet,  and  weight  10  pounds  per  linear  foot,  the  ends  being  sus- 
pended from  two  points  on  the  same  level.  If  /  be  the  span,  s 
the  length  of  the  cable,  and  z  a  length  of  the  cable  whose  weight 
equals  the  horizontal  tension,  the  solution  of  the  problem  leads 

to  the  transcendental  equation  s=  \e^'  — e  ^^f  s,or  inserting 
the  numerical  values, 

(10  loV 

^ ,.,  _  .^        .e^  -  i^fz  =  o 

is  the  equation  to  be  solved.     The  first  derivative  equation  is 

/   L°  _L°\         10/  L°  -i5\ 

f{z)  =  -[e^  -e  ^ j+  -[e^  +e   ^)  =  0, 

and  this  substituted  in  f{z)  shows  that  one  real  root  is  less  than 
about  20.  Assume  z^  =15,  then /(^'J =0.486  and /'(2',)= 0.206, 
whence  by  Newton's  rule  (Art.  4)  z^^  13  nearly.  Next  for 
z^  —  13, /{z^)  =  —  0.0298  and/'(^3)  =  0.322,  whence  ^3  =  13.1. 
Lastly  for  z^  =  13. 1  /(z^)  =0.0012  a.nd /'(z^)  =  0.3142,  whence 
s^  =  13.096,  which  is  a  sufficiently  close  approximation.  The 
horizontal  tension  in  the  given  catenary  is  hence  130.96  pounds.* 

*  Since  «'  —  e~  =2sinh0,  this  equation  may  be  written  ii9  —  losinhS, 
where  6  =  103-1,  and  the  solution  may  be  expedited  by  the  help  of  tables  of 
hyperbolic  functions.     See  Chapter  IV. 


ALGEBRAIC    SOLUTIONS.  15 

Prob.  9.  Show  that  the  equation  3  sin  x  —  2X  —  5  =  0  has  but 
one  real  root,  and  compute  its  value. 

Prob.  10.  Find  the  number  of  real  roots  of  the  equation 
20:  +  log  X  —10  000  =  o,  and  show  that  the  value  of  one  of  them  is 
X  —  4995- 74- 

Art.  8.    Algebraic  Solutions. 

Algebraic  solutions  of  complete  algebraic  equations  are 
only  possible  when  the  degree  «  is  less  than  5.  It  frequently 
happens,  moreover,  that  the  algebraic  solution  cannot  be  used 
to  determine  numerical  values  of  the  roots  as  the  formulas 
expressing  them  are  in  irreducible  imaginary  form.  Neverthe- 
less the  algebraic  solutions  of  quadratic,  cubic,  and  quartic 
equations  are  of  great  practical  value,  and  the  theory  of  the 
subject  is  of  the  highest  importance,  having  given  rise  in  fact 
to  a  large  part  of  modern  algebra.  ,, 

The  solution  of  the  quadratic  has  been  known  from  very 
early  times,  and  solutions  of  the  cubic  and  quartic  equations 
were  effected  in  the  sixteenth  century.  A  complete  investiga- 
tion of  the  fundamental  principles  of  these  solutions  was,  how- 
ever, first  given  by  Lagrange  in  1770.*  This  discussion  showed, 
if  the  general  equation  of  the  w'*'  degree,  y(;ir)  =0,  be  deprived 
of  its  second  term,  thus  giving  the  equation /"(j)  =:  o,  that  the 
expression  for  the  root  y  is  given  by 

in  which  n  is  the  degree  of  the  given  equation,  go  is,  in  suc- 
cession, each  of  the  71^"^  roots  of  unity,  i,  e,  e',  .  .  .  e""',  and 
jj,  ^2,  .  .  .  ^„_,  are  the  so-called  elements  which  in  soluble  cases 
are  determined  by  an  equation  of  the  n  —  \^^  degree.  For 
instance,  if  w  =  3  the  equation  is  of  the  third  degree  era  cubic, 
the  three  values  of  od  are 
<»,  =  I,     00=  —\^^^/ —  I  =  e,     cl)  =  —  ^  — 1-/— 3  =  e', 

*  Memoirs  of  Berlin  Academy,  1769  and  1770:  reprinted  in  CEuvres  de 
Lagrange  (I'aris,  1868),  Vol.  H,  pp.  539-562,  See  also  Traite  de  la  resolution 
des  Equations  numeriques,  Paris,  1798  and  1808. 


16  THE    SOLUTION    OF    EQUATIONS. 

and  the  three  roots  are  expressed  by 

in  which  5/  and  jj'  are   found  to  be  the   roots  of  a  quadratic 
equation  (Art.  9). 

The  n  values  of  go  are  the  n  roots  of  the  binomial  equation 
00"  —  1=0.  If  «  be  odd,  one  of  these  is  real  and  the  others 
are  imaginary  ;  if  n  be  even,  two  are  real  and  n  —  2  are  imagi- 
nary.* Thus  the  roots  oi  oa*  —  i  =  o  are  -{-  i  and  —  i ;  those 
of  a/  —  1  =  0  are  given  above ;  those  of  go*  —  1=0  are 
-\-  1,  -{-  i,  —  I,  and  —  t  where  i  isy  —  i.  For  the  equation 
co'  —  1=0  the  real  root  is  -J-  i.  and  the  imaginary  roots  are 
denoted  by  e,  e',  e^  e";  to  find  these  let  09"  —  i  =0  be  divided 
by  GO—  I,  giving 

GO*  -{-  GO^  -\-  GO^  -\-  GO  -{-  I  ^  O, 

which  being  a  reciprocal  equation  can  be  reduced  to  a  quad- 
ratic, and  the  solution  of  this  furnishes  the  four  values, 

=  _^(i_4/5  +  |/_io-2  V5),    e^  =  -i(i+  V5  +  i/_  10  +  2Vl)> 
=  _j(i_4/5_   V-10-2VJ),     e'  =  -i(i+ VJ— V_io  +  2V5), 

where  it  will  be  seen  that  e.e*  =  i  and  e*.e'  =  i,  as  should  be 
the  case,  since  e'  =  i. 

In  order  to  solve  a  quadratic  equation  by  this  general 
method  let  it  be  of  the  form 

x^  -{-  2ax  -\-b  -=0, 

and  let  x  be  replaced  hy  y  —  a,  thus  reducing  it  to 

f  -{a"  -b)  =  o. 

Now  the  two  roots  of  this  are  y^=  -\-  s,  and  j,  =  —  s^,  whence 
the  product  of  ( j  —  s^  and  {y  +  s^  is 

f  -  s"  =  o. 

Thus  the  value  of  /  is  given  by  an  equation  of  the  first  degree, 

*  The  values  of  a?  are,  in  short,  those  of  the  n  "  vectors  "  drawn  from  the 
center  which  divide  a  circle  of  radius  unity  into  «  equal  parts,  the  first  vector 
QJi  =  I  being  measured  on  the  axis  of  real  quantities.     See  Chapter  X. 


e 


€* 


THE    CUBIC    EQUATION.  17 

s^  ^  a^  —  b;  and    since  x  =  — ^+J»  ^he  roots   of   the  given 
equation  are 

x^  =  —  a-\-  -yf  a'  —  b,  x^——a—  -r/a"  —  v^, 

which  is  the  algebraic  solution  of  the  quadratic. 

The  equation  of  the  n  —  i"'  degree  upon  which  the  solution 
of  the  equation  of  the  n^^  degree  depends  is  called  a  resolvent. 
If  such  a  resolvent  exists,  the  given  equation  is  algebraically 
solvable ;  but,  as  before  remarked,  this  is  only  the  case  for 
quadratic,  cubic,  and  quartic  equations. 

Prob.    II.    Show     that    the    six     6"^    roots    of    unity    are  +  i, 

+i(i+  ^/^z),  -Ki-  ^~^z),  -h  -Ki+  ^-3)>  -Ki-  ^^3)- 

Art.  9.    The  Cubic  Equation. 

All  methods  for  the  solution  of  the  cubic  equation  lead  to 

the  result  commonly  known  as  Cardan's  formula.*     Let  the 

cubic  be 

x^ -\- ^ax" -\- T,bx -}- 2c  =  o,  (i) 

and  let  the  second  term  be  removed  by  substituting/  —  a  for 

X,  giving  the  form, 

f -{- 3By -{- 2C  =  o,  (!') 

in  which  the  values  of  B  and  C  are 

B=  -a'-\-b,         C  =  a'  -  ^ab  -\-  c.  (2) 

Now  by  the  Lagrangian  method  of  Art.  8  the  values  of  y  are 

J'l  =  -^1  +  -y. '         y-.  =  e^.  +  e'-y^ .         J's  =  e'-^i  +  es, , 
in  which    e    and  e^   are   the    imaginary   cube    roots    of  unity. 
Forming  the  products   of  the   roots,  and   remembering  that 
e'  =  I  and  e'  -[-  e  -j-  i  =  o,  there  are  found 

y.y,y^  =  s;-^s;=  -2C. 

For  the  determination  of  s^  and  s^  there  are  hence  two  equa- 
tions from  which  results  the  quadratic  resolvent 
/-}-  2Cs^  —  B^  =  o,  and  thus 

s,  =  {-c+V^^T^O^   s,  =  {- c  -  VB^T^-y.    (3) 

*  Deduced  by  Ferreo  in  1515,  and  first  published  by  Cardan  in  1545. 


18  THE    SOLUTION    OF    EQUATIONS. 

One  of  the  roots  of  the  cubic  in  j/  therefore  is 

^.  =  (- ^ + y^T^O*  +  (- ^  -  \/^M^^)*, 

and  this  is  the  well-known  formula  of  Cardan. 

The  algebraic  solution  of  the  cubic  equation  (i)  hence  con- 
sists in  finding  B  and  C  by  (2)  in  terms  of  the  given  coefificients, 
and  then  by  (3)  the  elements  ^,  and  s^  are  determined.     Finally, 

x,  =  —  a-^  {s,-Y  -yj, 

x^=  -a-\{s,^s:)^\y/  -  l(s,— s:^,  (4) 

x^=  —  a  —  \^{s,  -f  5,)  —  h^/  —  sC-^i  —  s^, 

which  are  the  algebraic  expressions  of  the  three  roots. 

When  B^ -\- C^  is  negative  the  numerical  solution  of  the 
cubic  is  not  possible  by  these  formulas,  as  then  both  ^,  and  s^ 
are  in  irreducible  imaginary  form.  This,  as  is  well  known,  is 
the  case  of  three  real  roots,  s^  -\-  s^  being  a  real,  while  s^  —  s^  is 
a  pure  imaginary.*  When  B^  -f-  (7Ms  o  the  elements  5,  and  s^ 
are  equal,  and  there  are  two  equal  roots,  x^=.  x^^^  —  a-\-  C'^, 
while  the  other  root  is  ;trj  =  —  a  —  2Ck 

When  ^^ -j-  C^  is  positive  the  equation  has  one  real  and 
two  imaginary  roots,  and  formulas  (2),  (3),  and  (4)  furnish  the 
numerical  values  of  the  roots  of  (i).     For  example,  take  the 

cubic 

x'  —  4.5;tr'  +  I2jr  —  5  =  o, 

whence  by  comparison  with  (i)  are  found  a  =  —  !.$,  d  =  -\-  4, 
c  —  —2.5.  Then  from  (2)'are  computed  B  =  1.75,  C=-\-'^.i2^. 
These  values  inserted  in  (3)  give  s^  =  -f-0.9142,  s^  =  —  1.9142; 
thus  .y,  +  5,  =  —  i.o  and  s,—  s,  =  -\-  2.8284.     Finally,  from  (4) 

X,  =  1.5  —  1.0=  +0.5, 


x,=  1.5  +0.5  +  1.4142  4/  -  3  =  2  +  2.4495/, 


X,  =  1.5  +  0.5  —  1.4142  4/—  3  =  2  -  2.4495?, 
which  are  the  three  roots  of  the  given  cubic. 

.  *  The  numerical  solution  of  this  case  1.,  [iossible  whenever  the  angle  whose 


cosine  is  —  C/  ^  —  B^  can  be  geometrically  trisected. 


THE    QUARTIC    EQUATION,  19 

Prob.  12.  Compute  the  roots  of  x^  —  2jc  — 5=  o.  Also  the  roots 
of  x"^  +  o. 6.v'  —  5.76X  +  4.32  =  o. 

Prob.  13.  A  cone  has  its  altitude  6  inches  and  the  diameter  of 
its  base  5  inches.  It  is  placed  with  vertex  downwards  and  one  fifth 
of  its  volume  is  filled  with  water.  If  a  sphere  4  inches  in  diameter 
be  then  put  into  the  cone,  what  part  of  its  radius  is  immersed  in  the 
water  ?     (Ans.  0.5459  inches). 

Art.  10.    The  Quartic  Equation. 

The  quartic  equation  was  first  solved  in  1545  by  Ferrari, 
who  separated  it  into  the  difference  of  two  squares.  Descartes 
in  1637  resolved  it  into  the  product  of  two  quadratic  factors, 
Tschirnhausen  in  1683  removed  the  second  and  fourth  terms. 
Euler  in  1732  and  Lagrange  in  1767  effected  solutions  by 
assuming  the  form  of  the  roots.  All  these  methods  lead  to 
cubic  resolvents,  the  roots  of  which  are  first  to  be  found  in 
order  to  determine  those  of  the  quartic. 

The  methods  of  Euler  and  Lagrange,  which  are  closely 
similar,  first  reduce  the  quartic  to  one  lacking  the  second  term, 

/^6/?/+46>  +  Z)  =  o; 

and  the  general  form  of  the  roots  being  taken  as 

7i  =  +  ^^  +  ^^.  +  ^^3.         f.  =  -^s',  +  ^.-  '^^„ 

the  values  s^,  s^,  s^,  are  shown  to  be  the  roots  of  the  resolvent, 

s'  +  sBs'  +  iigB'  -  D)s  -  \0  =  o. 

Thus  the  roots  of  the  quartic  are  algebraically  expressed  in 
terms  of  the  coefficients  of  the  quartic,  since  the  resolvent  is 
solvable  by  the  process  of  Art.  9. 

Whatever  method  of  solution  be  followed,  the  following 
final  formulas,  deduced  by  the  author  in  1892,  will  result.* 
Let  the  complete  quartic  equation  be  written  in  the  form 

x'  -f  4^^"  +  6<^^''  +  4CX  -\-  {^  =  o.  (i) 

*  See  American  Journal  Mathematics,  1892,  Vol.  XIV,  pp.  237-245. 


20  THE    SOLUTION    OF    EQUATIONS. 

First,  let^,  h,  and  k  be  determined  from 

g^a^  -b,     h^b'-^c'  -  2abc  ■\- dg,     k=  ^ac  -b^  -  \d.   (2> 

Secondly,  let  /  be  obtained  by 

/  =  \{li  +  l/F+TO'  ^Uh-  s/lF^'f  (3) 

Thirdly,  let  u,  v,  and  w  be  found  from 

u=g-\-l,     v  =  2g-  I,     w  =  4m'  +  3^  —  12^^-         (4) 
Then  the  four  roots  of  the  quartic  equation  are 


x^=  —  a  -\-  Vu  —  ^  V  -\-  Vw, 
x^=  —  a  —  Vu  -{-  yv  —  Via, 


^ 


1-  (5) 


x^-=  —  a 


Vu  —  Vy  —  \/^^  ^ 

in  which  the  signs  are  to  be  used  as  written  provided  that 
2a^  —  lab  +  ^  is  a  negative  number;  but  if  this  is  positive  all 
radicals  except  Vw  are  to  be  reversed  in  sign. 

These  formulas  not  only  serve  for  the  complete  theoretic 
discussion  of  the  quartic  (i),  but  they  enable  numerical  solu- 
tions to  be  made  whenever  (3)  can  be  computed,  that  is,  when- 
ever 1^  -\- k^  is  positive.  For  this  case  the  quartic  has  two  real 
and  two  imaginary  roots.  If  there  be  either  four  real  roots  or 
four  imaginary  roots  k^  -\-  k^  is  negative,  and  the  irreducible 
case  arises  where  convenient  numerical  values  cannot  be  ob- 
tained, although  they  are  correctly  represented  by  the  formulas. 

As  an  example  let  a  given  rectangle  have  the  sides/  and  q, 
and  let  it  be  required  to  find  the  length  of  an  inscribed  rec- 
tangle whose  width  is  m.  If  x  be  this  length,  this  is  a  root  of 
the  quartic  equation 

x'  -  iP'  +  ^'  +  2m')x'  4-  4pqmx  -  {p"  +  ^'  -  myn^  =  O, 

and  thus  the  problem  is  numerically  solvable  by  the  above 
formulas  if  two  roots  are  real  and  two  imaginary.  As  a  special 
case  let/  =  4  feet,  ^  =  3  feet,  and  m  =  i  foot ;  then 

X*  —  2'JX''  -\-  \Zx  —  24  =  O. 


QUINTIC    EQUATIONS.  21 

By  comparison  with  (i)  are  found  a  =  o,  i>  =  —  4^,  c  =  -}-  I2, 
and  ^=—24.  Then  from  (2),  £  =  -{-4^,  h=—^i-,  and 
J5=z  -\-  -Y-.  Thus  /^'  +  k"  is  positive,  and  from  (3)  the  value  of  I 
is— 3.6067.  From  (4)  are  now  found,  ?^  =  -1-0.8933,  ^'=  12.6067, 
and  w  =  -j-  161.20.  Then,  since  c  is  positive,  the  values  of  the 
four  roots  are,  by  (5), 


*-,  =  —  0.945  —  1^12.607  -\-  12.697  =  —  5.975  feet, 
;r,  =  —  0.945  -h  i/i 2.607  -1-  12.697  =  -\-  4.085  feet, 
jfj  =  +  0.945  —  1/12.607  —  12.697  =  -f  0.945  —  0.30/, 
X,  =  -h  O  945  +  1^12.607-  12.697  =  +  0.945  +  0.30/, 

the  second  of  which  is  evidently  the  required  length.     Each  of 

these  roots  closely  satisfies  the  given  equation,  the  slight  dis- 

<:repancy  in  each  case  being  due  to  the  rounding  off  at  the  third 

decimal.* 

Prob.  14.  Compute  the  roots  of  the  equation  jc*  +  7>^  +  6  =  o. 
(Ans.  —  1.388,  —  1. 000,  1. 194  ±  1. 701/.) 


Art.  11.    QuiNTic  Equations. 

The  complete  equation  of  the  fifth  degree  is  not  algebraic- 
ally solvable,  nor  is  it  reducible  to  a  solvable  form.  Let  the 
equation  be 

x"  +  S^-*"'  +  5^^'  +  5^-^"  +  5^^-^  +  2^  =  o, 
and  by  substituting  j/  —  a  for  x  let  it  be  reduced  to 

/-{-  S^/  +  SC/  +  sBf-\-2E  =  0. 
The  five  roots  of  this  are,  according  to  Art.  8, 

J.  =  €S^  4-  e's,  -\-  e's,  4-  eV„ 
Js  =  e's,  +  eV,  +  es,  -[-  e's,, 
y*  =  e'-y.  +  es,  -h  6^3  +  e\, 

in  which  e,  e*,  e'  e*  are  the  imaginary  fifth  roots  of  unity.     Now 
if  the  several  products  of  these  roots  be  taken  there  will  be 

*  This  example  is  known  by  civil  engineers  as  the  problem  of  finding  the 
length  of  a  strut  in  a  panel  of  the  Howe  truss. 


22  THE    SOLUTION    OF    EQUATIONS, 

found,  by  (4)  of  Art.  6,  four  equations  connecting  the  four  ele- 
ments s^,  s^,  s^,  and  s^,  namely, 

—  B  —  s,s,  +  V3. 

—  C  =  s;'s,  +  s,%  4-  s^'s,  +  sX, 

-2E=  s; + s: + ,^3^ + s:  +  5(^;^.v,  ^-  ^^,.3^. + ^;^;^3+ ^3'^;^:) 

but  the  solution  of  these  leads  to  an  equation  of  the  I20th 
degree  for  s,  or  of  the  24th  degree  for  s".  However,  by  taking 
SjS^  —  s^s^  or  s^^  -{-  ,$•/  +  s^^  -\-  s^^  as  the  unknown  quantity,  a 
resolvent  of  the  6th  degree  is  obtained,  and  all  efforts  to  find 
a  resolvent  of  the  fourth  degree  have  proved  unavailing. 

Another  line  of  attack  upon  the  quintic  is  in  attempting  to 
remove  all  the  terms  intermediate  between  the  first  and  the 
last.  By  substituting  y  +/J  4-^  for  x,  the  values  of/  and  g 
may  be  determined  so  as  to  remove  the  second  and  third  terms 
by  a  quadratic  equation,  or  the  second  and  third  by  a  cubic 
equation,  or  the  second  and  fourth  by  a  quartic  equation,  as 
was  first  shown  by  Tschirnhausen  in  1683.  By  substituting 
y  ~\~py  ~\~  ^y  ~h  ^  fo''  -^T  three  terms  may  be  removed,  as  was 
shown  by  Bring  in  1786.  By  substituting  y-|-/j'''+ ^y'+^J+^ 
for  X  it  was  thought  by  Jerrard  in  1833  that  four  terms  might 
be  removed,  but  Hamilton  showed  later  that  this  leads  to 
equations  of  a  degree  higher  than  the  fourth. 

In  1826  Abel  gave  a  demonstration  that  the  algebraic  solu- 
tion of  the  general  quintic  is  impossible,  and  later  Galois 
published  a  more  extended  investigation  leading  to  the  same 
conclusion.*  The  reason  for  the  algebraic  solvability  of  the 
quartic  equation  may  be  briefly  stated  as  the  fact  tliat  thtre 
exist  rational  three-valued  functions  of  four  quantities.  There 
are,  however,  no  rational  four-valued  functions  of  five  quan- 
tities, and  accordingly  a  quartic  resolvent  cannot  be  found  for 
the  general  quintic  equation. 

*  Jordan's  Trait6  des  substitutions  et  des  Equations  algebriques;  Paris,  1870. 
Abhandlungen  iiber  die  algebraische  Auflosung  der  Gleichungen  von  N.  H. 
Abel  und  Galois;  Berlin,  1889. 


QUINTIC    EQUATIONS.  23 

There  are,  however,  numerous  special  forms  of  the  quintic 
whose  algebraic  solution  is  possible.  The  oldest  of  these  is  the 
quintic  of  De  Moivre, 

which  is  solved  at  once  by  making  s^^  s^^o  in  the  element 
equations  ;  then  —  B  ~  s^s^  and  —  2E  =  s^  -\-  s^",  from  which 
J,  and  s^  are  found,  and  j,  =  ^^  -)-j^,  or 

while  the  other  roots  are  j,  =  es,  -J-  e*s^ ,  y^  =  eV,  -f-  e\, 
y^=.  eV,  +  e\ ,  and  y^  =  eV,  +  e^^ .  If  ^'  -j-  £""  be  negative, 
this  quintic  has  five  real  roots;  if  positive,  there  are  one  real 
and  four  imaginary  roots. 

When  any  relation,  other  than  those  expressed  by  the  four 
element  equations,  exists  between  s^,  s^,  s^,  s^,  the  quintic  is 
solvable  algebraically.  As  an  infinite  number  of  such  relations 
may  be  stated,  it  follows  that  there  are  an  infinite  number  of 
solvable  quintics.  In  each  case  of  this  kind,  however,  the  co- 
efficients of  the  quintic  are  also  related  to  each  other  by  a 
certain  equation  of  condition. 

The  complete  solution  of  the  quintic  in  terms  of  one  of  the 
roots  of  its  resolvent  sextic  was  made  by  McClintock  in  1884.* 
By  this  method  s^,  s^\  s^\  and  5/  are  expressed  as  the  roots  of 
a  quartic  in  terms  of  a  quantity  /  which  is  the  root  of  a  sextic 
whose  coefficients  are  rational  functions  of  those  of  the  given 
quintic.  Although  this  has  great  theoretic  interest,  it  is,  of 
course,  of  little  practical  value  for  the  determination  of  numer- 
ical values  of  the  roots. 

By  means  of  elliptic  functions  the  complete  quintic  can, 
however,  be  solved,  as  was  first  shown  by  Hermite  in  1858. 
For  this  purpose  the  quintic  is  reduced  by  Jerrard's  transfor- 
mation to  the  form  x'  -\-  ^dx -\-  2e  =  o,  and  to  this  form  can 
also  be  reduced  the  elliptic  modular  equation  of  the  sixth 
degree.     Other  solutions  by  elliptic  functions  were  made  by 

*  American  Journal  of  Mathematics,  iS86,  Vol.  VIII,  pp.  49-83. 


24  THE   SOLUTION    OF    EQUATIONS. 

Kronecker  in  1861  and  by  Klein  in  1884  *  These  methods, 
though  feasible  by  the  help  of  tables,  have  not  yet  been  sys- 
tematized  so  as  to  be  of  practical  advantage  in  the  numerical 
computation  of  roots. 

Prob.  15.  If  the  relation  j'jj'^  ^i'^^,  exists  between  the  elementg 
show  that  s;  +  s,'  +  ^3'  +  ^/  =  -  2^. 

Prob.  16.  Compute  the  roots  of  y  +  10^+  2oy  +  6  =  0,  and 
also  those  oiy  —  lo^  +  207  +  6  =  0. 

Art.  12.    Trigonometric  Solutions. 

When  a  cubic  equation  has  three  real  roots  the  most  con- 
venient practical  method  of  solution  is  by  the  use  of  a  table  of 
sines  and  cosines.  If  the  cubic  be  stated  in  the  form  (i)  of 
Art.  9,  let  the  second  term  be  removed,  giving 

Now  suppose  J  =  2r  sin   6,  th,en  this  equation  becomes 

B  C 

8  sin' ^+6-,  sin  ^+2-3  =  0, 

and  by  comparison  with  the  known  trigonometric  formula 

8  sin'  /9  —  6  sin  ^  +  2  sin  3(9  =  O, 
there  are  found  for  r  atid  sin  3^  the  values 


y=t^_  B,  sin  3^  =  C/  V-  B\ 

in  which  B  is  always  negative  for  the  case  of  three  real  roots 
(Art.  9).  Now  sin  36*  being  computed,  36^  is  found  from  a  table 
of  sines,  and  then  B  is  known.     Thus, 

J,  =  2r  sin  d,  y^  —  2r  sin  (120°  +  B),  /,  =  2r  sin  (240°  +  (f), 

are  the  real  roots  of  the  cubic  in  y.\ 

*  For  an  outline  of  these  transcendental  methods,  see  Hagen's  Synopsis  der 
hoheren  Mathematik,  Vol.  I,  pp.  339-344. 

f  When  B^  is  negative  and  numerically  less  than  CS  as  also  when  B^  is 
positive,  this  solution  fails,  as  then  one  root  is  real  and  two  are  im.aginary.  In 
this  case,  however,  a  similar  method  of  solution  by  means  of  "hyperbolic  sines 
is  possible.  See  Grunert's  Archiv  fiir  Mathematik  und  Physik,  Vol.  xxxviii, 
pp.  48-76. 


TRIGONOMETRIC    SOLUTIONS.  25 

For  example,  the  depth  of  flotation  of  a  sphere  whose  diam- 
eter is  2  feet  and  specific  gravity  0.65,  is  given  by  the  cubic 
equation  x'  —  s^""  -{-  2.6  =  o  (Art.  6).  Placing  x  =  y  -{-  i  this 
reduces  tojj/'—  37  -|-0.6  =  o,  for  which  B  =  —  i  and  6' =+0.3. 
Thus  r=  I  and  sin  36*  z=-|-o.3.  Next  from  a  table  of  sines, 
^0  =  17°  27',  and  accordingly  0  =  S°  49'.     Then 

/,  =  2  sin  5°  49'  =  +0-2027, 
^,  =  2  sin  125°  49' =  + 1-6218, 
^,  =  2  sin  245°  49'  =  —  1.8245. 

Adding  i  to  each  of  these,  the  values  of  x  are 

x^=  -\-  1.203  f^^t,     x,  =  -{-  2.622  feet,     x^  =—0.825  feet ; 

and  evidently,  from  the  physical  aspect  of  the  question,  the 
first  of  these  is  the  required  depth.  It  may  be  noted  that  the 
number  0.3  is  also  the  sine  of  162°  11',  but  by  using  this  the 
three  roots  have  the  same  values  in  a  different  order. 

When  the  quartic  equation  has  four  real  roots  its  cubic  re- 
solvent has  also  three  real  roots.  In  this  case  the  formulas  of 
Art.  10  will  furnish  the  solution  if  the  three  values  of  /  be  ob- 
tained from  (3)  by  the  help  of  a  table  of  sines.  The  quartic 
being  given,  ^,  /i,  and  k  are  found  as  before,  and  the  value  of 
^  will  always  be  negative  for  four  real  roots.     Then 


r  =  V—  k,  sin  3^  =  —  /i/r\ 

and  36*  is  taken  from  a  table  ;  thus  6  is  known,  and  the  three 
values  of  /are 

/,  =  r  sin  d,         l,  =  r  sin(i20°  +  6),         I,  =  r  sin  (240°  +  8). 

Next  the  three  values  of  u,  of  7>,  and  of  w  are  computed,  and 
those  selected  which  give  u,  w,  and  v  —  Vzv  all  positive  quanti- 
ties.    Then  (5)  gives  the  required  roots  of  the  quartic. 

As  an  example,  take  the  case  of  the  inscribed  rectangle  in 
Art.  10,  and  let  /  =  4  feet,  ^  =  3  feet,  m  =  Vi^  feet;  then  the 
quartic  equation  is 

AT*  —  5  !;»:•  +  48  i^Fs  .^  —  1 56  =  O. 


26  THE    SOLUTION    OF   EQUATIONS. 

Here  a  =  o,  !f  =  —  S^,  c  =  -{-  12  ViS,  and  d  =  —  156.  Next 
g  =-\-  Si,  /i=  —  -§-1^,  and  k  =—  ^'  The  trigononnetric  work 
now  begins;  the  value  of  r  is  found  to  be  +  4^,  and  that  of 
sins^'  to  be  +  0.7476;  hence  from  the  table  $6  =  48°  23',  and 
6=  16°  07' 40".  The  three  values  of  /  are  then  computed 
by  logarithmic  tables,  and  found  to  be, 

/.  =  +  1.250,  /,  =  +  3.1187,  /,  =  -4.3687. 

Next  the  values  of  u,  v,  and  w  are  obtained,  and  it  is  seen  that 
only  those  corresponding  to  /,  will  render  all  quantities  under 
the  radicals  positive  ;  these  quantities  are  ti  =  9.75,  v  =  15.75, 
and  zv  =  192.0.     Then  the  four  roots  of  the  quartic  are 

;r.  = -8.564,     X,  =+  2.319,     x,  =  -{-  1.746,     X,  =  +  4.499  feet, 

of  which  only  the  second  and  third  belong  to  inscribed  rec- 
tangles, while  the  first  and  fourth  belong  to  rectangles  whose 
corners  are  on  the  sides  of  the  given  rectangle  produced. 

Trigonometric  solutions  of  the  quintic  equation  are  not 
possible  except  for  the  binomial  x^  ±  a,  and  the  quintic  of 
De  Moivre.  The  general  trigonometric  expression  for  the  root 
»of  a  quintic  lacking  its  second  term  isjj/=2r,  cos^,+2r5  cos  ^, , 
and  to  render  a  solution  possible,  r,  and  r^ ,  as  well  as  cos  ^, 
and  cos  6^2'  must  be  found;  but  these  in  general  are  roots  of 
equations  of  the  sixth  or  twelfth  degree  :  in  fact  r,"  is  the  same 
as  the  function  s^s^  of  Art.  11,  and  r/  is  the  same  as  s^s^. 
Here  cos^,  and  cos^',  may  be  either  circular  or  hyperbolic 
cosines,  depending  upon  the  signs  and  values  of  the  coefificients 
of  the  quintic. 

Trigonometric  solutions  are  possible  for  any  binomial  equa- 
tion, and  also  for  any  equation  which  expresses  the  division  of 
an  angle  into  equal  parts.  Thus  the  roots  of  ;ir^  +  i  =0  are 
cosm  30°  ±  i  sin  7n  30°,  in  which  m  has  the  values  i,  2,  and  3. 
The  roots  of  x''—Sx^-\-c^x  —  2  cos 5  (9  =  o  are  2  cos  {m  y2°-\-H) 
where  w  has  the  values  o,  1,2,  3,  and  4. 

Prob.  17.  Compute  by  a  trigonometric  solution  the  four  roots  of 
the  quartic^'  +  4^^"  —  240:^  — 76;*;— 29  =  o.  (Ans.  —6.734,  —1-550, 
+  0.262,  +  4.022). 


REAL    ROOTS    BY    SERIES.  27 

Prob.  1 8.  Give  a  trigonometric  solution  of  the  quintic  equation 
x^  —  ^Lx^  4"  S^^^  —  2(?  =  o  for  the  case  of  five  real  roots.  Compute 
the  roots  when  d  =i  and  e  :=  0.752798.  (Ans.  —1.7940,  —  1.3952, 
0.2864,  0-9317.  I-97IO-) 

Art.  13.    Real  Roots  by  Series. 

The  value  of  x  in  any  algebraic  equation  may  be  expressed 
as  an  infinite  series.  Let  the  equation  be  of  any  degree,  and 
by  dividing  by  the  coefficient  of  the  term  containing  the  first 
power  of  X  let  it  be  placed  in  the  form 

a=zx^dx'  +  ex '+  dx'  +  ex'  4-/r'  + .  .  . 

Now  let  it  be  assumed  that  x  can  be  expressed  by  the  series 

X  =  a  -\-  jna"  -\-  nc^  -\-p^*  +  ^'^^  +  •  •  • 

By  inserting  this  value  of  x  in  the  equation  and  equating  the 
coefficients  of  like  powers  of  a,  the  values  of  w,  ?i,  etc.,  are 
found,  and  then 

-(42^'-  84<^V+  28^V+  28<^^'-  T^e-  ycd -^ /)a' -\- .  .  ., 

is  an  expression  of  one  of  the  roots  of  the  equation.  In  order 
that  this  series  may  converge  rapidly  it  is  necessary  that  a 
should  be  a  small  fraction.* 

To  apply  this  to  a  cubic  equation  the  coefficients  ^,  e,/,  etc., 
are  made  equal  to  O,  For  example,  let  x^  —  t,x -}- 0.6  =  o  ; 
this  reduced  to  the  given  form  is  0.2  =  x  —  ^x^,  hence  a  =  0.2, 
i  =  O,  e  =^  —  ^,  and  then 

x  =  o.2-\-i.  0.2'  +  i .  0.2*  +  etc.  =  +  0.20277, 

which  is  the  value  of  one  of  the  roots  correct  to  the  fourth 
decimal  place.  This  equation  has  three  real  roots,  but  the 
series  gives  only  one  of  them  ;  the  others  can,  however,  be 
found  if  their  approximate  values  are  known.  Thus,  one  root 
is  about  -{-1.6,  and  by  placing  x=j'-{-i.6  there  results  an 
equation  in  j;/ whose  root  by  the  series  is  found  to  be -f- 0.02 18, 
and  hence  -j-  1.6218  is  another  root  of  x^  —  ^x-\-o.6  =  o. 

*This  method  is  given  by  J.  B.  Molt  in  The  Analyst,  1882,  Vol.  IX,  p.  104. 


28  THE    SOLUTION    OF    EQUATIONS. 

Cardan's  expression  for  the  root  of  a  cubic  equation  can  be 
expressed  as  a  series  by  developing  each  of  the  cube  roots  by 
the  binomial  formula  and  adding  the  results.  Let  the  equa- 
tion he  y  -{-  ^By  -\-  2C  =  o,  whose  root  is,  by  Art.  9, 

J  =  (-  C+VB-'^Cy+i-  C-  VB'  +  C)\ 
then  this  development  gives  the  series, 

—    (—r\h{    —  -    .  .  ^-  5  •  8    ,      2  ■  5  .  8  .  1 1  .  14   ,_         \ 
•^  ~    ^         ^\         2  2.3.4  2.3.4.5,6    ^       •••/' 

in  which  r  represents  the  quantity  {B^ -\-  C^^/^C^.  If  r  =  o 
the  equation  has  two  equal  roots  and  the  third  root  is  2(  — 6~/. 
If  r  is  numerically  greater  than  unity  the  series  is  divergent, 
and  the  solution  fails.  If  r  is  numerically  less  than  unity  and 
sufficiently  small  to  make  a  quick  convergence,  the  series  will 
serve  for  the  computation  of  one  real  root.  For  example,  take 
the  equation  x^  —  6x  -\-6  -=.0,  where  B=-  —  2  and  C  =  3  ; 
hence  r  =  1/8 1,  and  one  root  is 

y  =  —  2.8845(1  —  0.01235  —  0.00051  —  0.00032—)  =  —  2.846, 

which  is  correct  to  the  third  decimal.  In  comparatively  few 
-cases,  however,  is  this  series  of  value  for  the  solution  of  cubics. 

Many  other  series  for  the  expression  of  the  roots  of  equa- 
tions, particularly  for  trinomial  equations,  have  been  devised. 
One  of  the  oldest  is  that  given  by  Lambert  in  1758,  whereby 
the  root  of  x"  -\^  ax  —  b  ^=  o  is  developed  in  terms  of  the 
ascending  powers  of  b/a.  Other  solutions  were  published  by 
Euler  and  Lagrange.  These  series  usually  give  but  one  root, 
and  this  only  when  the  values  of  the  coefificients  are  such  as  to 
render  convergence  rapid. 

Prob.  19.  Consult  Euler's  Anleitung  zur  Algebra  (St.  Petersburg, 
'77i)>  PP-  i43-i5o>^i^d  apply  his  method  of  series  to  the  solution  of 
a  quartic  equation. 

Art.  14.    Computation  of  all  Roots. 

A  comprehensive  and  valuable  method  for  the  solution  of 
equations  by  series  was  developed  by  McClintock,  in  1894,  by 


COMPUTATION    OF    ALL    ROOTS  29^ 

means  of  his  Calculus  of  Enlargement.^  By  this  method  all 
the  roots,  whether  real  or  imaginary,  may  be  computed  from  a 
single  series.  The  following  is  a  statement  of  the  method  as 
applied  to  trinomial  equations  : 

Let  ,r"  =  nAx"~''  -\-  B"   be  the   given   trinomial   equation. 
Substitute  x  =  By  and  thus  reduce  the  equation  to  the  form 
y  =  nay"'^ -{-I  where  a  =  A/B^.     Then  if  .5"  is  positive,  the 
roots  are  given  by  the  series 

y  =  Go-\- 00"-^  a  -\-  oo^-\\  —2k-\-  n)a'/2 ! 
4-(S>'-3*(i  -3/^+«)(i-3/^+2;/K/3 ! 

in  which  0,1  represents  in  succession  each  of  the  roots  of  unity. 
If,  however,  B"  is  negative,  the  given  equation  reduces  to 
y  ^  nay"~'' — I,  and  the  same  series  gives  the  roots  if  go  be 
taken  in  succession  as  each  of  the  roots  of  —  i. 

In  order  that  this  series  may  be  convergent  the  value  of  «• 
must  be  numerically  less  than  k~''{fi  —  kY~" ;  thus  for  the  quar- 
tic  y*  =  ^ax  -\-  i,  where  «  =  4  and  >^  =  3,  the  value  of  a  must 
be  less  than  27~*. 

To  apply  this  method  to  the  cubic  equation  x^^TfAx±_B^^ 
place  «  =  3  and  ^  =  2,  and  put  y  =  Bx.  It  then  becomes 
y  =  yzy  ±  I  where  a  =  A/B\  and  the  series  is 

in  which  the  values  to  be  taken  for  go  are  the  cube  roots  of  i 
or  —  I,  as  the  ease  may  be.  For  example,  let  x^—  2x  —  $  =0. 
Placing  jj/=  59^,  this  reduces  to  y =0.684 jf-f-i.  Here  «  =  o.228, 
and  as  this  is  less  than  4-*  the  series  is  convergent.  Making 
GO  =  i,  the  first  root  is 

y  =  I  -\-  0.2280  —  0.0039  +  0.0009  =  1.2250. 

*See  Bulletin  of  American  Mathematical  Society,  i8g4,  Vol.  I,  p.  3;  also- 
American  Journal  of  Mathematics,  1895,  Vol.  XVII,  pp.  89-110. 


30  THE   SOLUTION    OF    EQUATIONS. 

Next  making  co  =  —  i  +  i  V^,  go*  is  —  ^  — ^  V—  3, 
and  the  corresponding  root  is  found  to  be 

J  =  —  0.6125  +  0.3836  V—  3. 


Again,  making  &?  =  —  i  —  i  V—  3  the  third  root  is  found  to 
be  the  conjugate  imaginary  of  the  second.  Lastly,  multiplying 
each  value  of  jj/  by  5 J, 

X  =  2.095,  ;jr  =  -  1.047  ±  1-136  V—  I, 

which  are  very  nearly  the  roots  of  ;i:'  —  2.a:  —  5  =  O. 

In  a  similar  manner  the  cubic  x^-\-  2x  -\-$  =  o  reduces  to 
y  =  —  o.684>'  —I,  for  which  the  series  is  convergent.     Here 
the  three  values  of  oj  are,   in  succession,  —  I,  i  +i  V—  3, 
_  ^_|_  j4,/_  3^    and    the   three    roots   are  j  =— 0.777   and 
/  =  0.388  ±  1.137/. 

When  all  the  roots  are  real,  the  method  as  above  stated 
fails  because  the  series  is  divergent.  The  given  equation  can, 
however,  be  transformed  so  as  to  obtain  n  —  k  roots  by  one 
application  of  the  general  series  and  k  roots  by  another.  As 
an  example,  let  x^  —  2^t,x  +  330  =  o.  For  the  first  applica- 
tion  this  is  to  be  written  in  the  form 

x'     ,  330 
243  ^  243' 

for  which  «  =  I  and  k  =  —  2.     To  make  the  last  term  unity 

place  X  =  -^^jy*  and  the  equation  becomes 

whence  a  =  330*73.243'.  These  values  of  «,  k,  and  a  are  now 
inserted  in  the  above  general  value  of  jj/,  and  co  made  unity; 
thus  jj/=  0.9983,  whence  ;r,  =1.368  is  one  of  the  roots.  For 
♦;he  second  application  the  equation  is  to  be  written 

^ = -  M^^"' + 243. 
243 


ROOTS    OF   UNITY.  31 

for  which  «  =  2  and  >&  =  3.     Placing  x  =  243*;',  this  becomes 

y  =  -  345  -.  + ,, 

2435-^ 

whence  ^=  —  iio/243s  and  the  series  is  convergent.  These 
values  of  «,  k,  and  «  are  no  a  inserted  in  the  formula  for  ^, 
and  CO  is  made  -f-  i  and  —  i  in  succession,  thus  giving  two 
values  for  J/,  from  which  x^  =  14.86  and  ;r,  =  —  i6.22  are  the 
other  roots  of  the  given  cubic. 

McClintock  has  also  given  a  similar  and  more  general 
method  applicable  to  other  algebraic  equations  than  trinomials. 
The  equation  is  reduced  to  the  form  y"  =  na  .  ^yy  ±.  i,  where 
na .  cfyy  denotes  all  the  terms  except  the  first  and  the  last. 
Then  the  values  oi  y  are  expressed  by  the  series 

d  a" 

in  which  the  values  of  a>  are  to  be  taken  as  before.  The 
method  is  one  of  great  importance  in  the  theory  of  equations, 
as  it  enables  not  only  the  number  of  real  and  imaginary  roots 
to  be  determined,  but  also  gives  their  values  when  the  conver- 
gence of  the  series  is  secured. 

Prob.  20.  Compute  by  the  above  method  all  the  roots  of  the 
quartic  x*  -\-x-\-  10  =  0. 

Art.  15.    Roots  of  Unity. 

The  roots  of  +1  and  —  i  are  required  to  be  known  in  the 
numerical  solution  of  algebraic  equations  by  the  method  of  the 
last  article.  From  the  theory  of  binomial  equations  given  in  all 
text-books  on  algebra,  the  n  roots  of  +1  are 

(  +  1)"^  =cos  (m/n)27T+i  sin  (m/n)27:,     w  =  i,  2,  3, .  .  .  w,      (i) 
while  those  of  —  i  are  expressed  by 

(  —  i)  **  =  cos  {m/n)Tt-\-i  sin  (w/n);r,      w  =  i,  2,  3, . . .  «,       (2) 


32  THE    SOLUTION   OF    EQUATIONS. 

in  which  i  represents  the  square  root  of  —  i.  From  these  general 
formulas  it  is  seen  that  the  two  imaginary  cube  roots  of  + 1  are 

£i  =  -  ^  +  ^iVs  =  -0.5  +0. 86602  54i, 
e2=-^-^iVs=-o.S-o.S66o2S4h 
and  that  the  two  imaginary  cube  roots  of  —  i  are 

£/=  +i  +  hiVs=  +0.5+0.8660254/, 
£/=  +i-i/\/3==  +0.5-0.8660254/. 

For  the  first  case  £1+62  +  1=0  and  £i£2  =  i>  ^.s  also  £1  =  ^2^  and 
£^=£l,  and  similar  relations  apply  to  the  other  case. 

The  imaginary  fifth  roots  of  positive  unity  are  given  in  Art.  8 
expressed  in  radicals;  reducing  these  to  decimals,  or  deriving 
them  from  the  above  formula  (i)  with  the  help  of  a  trigonometric 
table,  there  result 

£  =+0.3090170+0.9510565/,  £^=—0.8090170+0.5877853/, 
£*=  +0.3090170— 0.9510565/,        £^=  —0.8090170—0.5877853/, 

while  the  imaginary  fifth  roots  of  negative  unity  are  obtained 
from  these  by  changing  the  signs.  In  general,  if  co  is  an  imaginary 
n^^  root  of  positive  unity,  —  a»  is  an  imaginary  n^^  root  of  nega- 
tive unity. 

The  imaginary  sixth  roots  of  positive  unity  may  be  expressed 
in  terms  of  the  cube  roots.  Let  £  be  one  of  the  imaginary  cube 
roots  of  +1,  then  the  imaginary  sixth  roots  of  +1  are  +£,  +£^, 
—  £,  —  £^;  these  are  also  the  imaginary  sixth  roots  of  —  i. 

From  (i)  the  imaginary  seventh  roots  of  +1  are  found  to  be 

£  =+0.6234898+0.7818316/,  £^=+0.6234898-0.7818316/, 
£^=  — 0.2225209 +0.9749234^',  £^=  —0.2225209—0.9749234/, 
£^=  -0.9009688+0.4338837/,     £^=  -0.9009688-0.4338837/, 

and  if  the  signs  of  these  be  reversed  there  result  the  imaginary 
seventh  roots  of  —  i. 

The  imaginary  eighth  roots  of  +1  are  +/,  — /,  +^\^2(i±i), 
and  —  |V2(i±/).    The  imaginary  ninth  roots  of  +1  are  the  two 


SOLUTIONS    BY    MACLAURIN'S    FORMULA.  33 

imaginary  cube  roots  of  + 1 ,  cos  -^n  ±  i  sin  ^n,  and  cos  f  tt  ±  f  sin  %7z. 
The  imaginary  tenth  roots  of  +i  are  the  five  imaginary  roots 
of  +1  and  the  five  imaginary  roots  of  —  i.  For  any  value  of 
n  the  roots  of  +i  may  be  graphically  represented  in  a  circle 
of  unit  radius  by  taking  one  radius  as  +i  and  drawing  other 
radii  to  divide  the  circle  into  n  equal  parts;  if  unit  distances 
normal  to  +i  and  —  i  be  called  -\-i  and  —i,  the  n  radii  repre- 
sent all  the  roots  of  + 1 .  When  this  figure  is  viewed  in  a  mirror, 
the  image  represents  the  n  roots  of  —  i.  Or,  in  other  words, 
the  {m/nY^  roots  of  +i  are  unit  vectors  which  make  the  angles 
{m/n)2n  with  the  unit  vector  +i,  while  the  {m/nY^  roots  of  —  i 
are  unit  vectors  which  make  the  angles  {m/n)27t  with  the  unit 
vector  —I. 

The  n  roots  of  any  unit  vector  cos  ^+i  sin  (9  are  readily  found 
from  De  Moivre's  theorem  by  the  help  of  trigonometric  tables. 
Accordingly  the  cube  roots  of  this  vector  are  cos  \0-\-i  sin  \d, 
co?>\{d-\-27t)-\-i?>m.^{d  +  27i)  and  cos-J(^+47r)+isin^((9+4;r);  the 
vectors  representing  these  three  roots  divide  the  circle  into  three 
parts.  The  trigonometric  solution  of  the  cubic  equation  (Art.  12) 
is  one  application  of  De  Moivre's  theorem. 

Prob.  21.  Compute  to  six  decimal  places  two  or  more  of  the  eleventh 
imaginary  roots  of  unity. 

Prob.  22.  Compute  to  five  decimal  places  the  five  roots  of  the 
equation  .v^— 0.8  — o.6i=o. 

Prob.  23.  Compute  to  five  decimal  places  the  six  roots  of  the 
equation  x^—2>o-\-6oi=Q. 


Art.  16.     Solutions  by  Maclaurin's  Formula. 

In  1903  Lambert  pubhshed  a  method  for  the  expression  by 
Maclaurin's  formula  of  the  roots  of  equations  in  infinite  series.* 
It  applies  to  both  algebraic  and  transcendental  equations,  and 
for  the  former  it  gives  all  the  roots  whether  they  be  real  or  imag- 
inary.    The  method  is  based  on  the  device  of  introducing  a 

*  Proceedings  American  Philosophical  Society,  Vol.  42.        > 


34  THE    SOLUTION   OF    EQUATIONS. 

factor  X  into  all  the  terms  but  two  of  the  equation /"(;y)  =o,  whereby 
y  becomes  an  implicit  function  of  x.  The  successive  derivatives 
of  y  with  respect  to  x  are  then  obtained,  and  their  values,  as  also 
those  of  y,  are  evaluated  for  x  =  o.  By  Maclaurin's  formula, 
the  expansions  of  y  in  powers  of  x  become  known,  and  if  x  be 
made  unity  in  these  expansions,  the  roots  of /(y)=o  are  found, 
pEOvided  the  resulting  series  are  convergent. 

To  illustrate  this  method  by  a  numerical  example,  take  the 
quartic  equation 

^y^—sy^  +  lsy  —  T-o  000=0,  (i) 

and  introduce  an  x  into  the  second  and  third  terms,  thus, 

y^  —  Sxy^  +  y^xy  — 10000=0.  (2) 

By  Maclaurin's  formula  y  may  be  expressed  in  terms  of  x, 
and  then  when  x  is  made  unity,  the  four  series  thus  obtained 
furnish  the  four  roots  of  (i).     Maclaurin's  formula  is  ' 


/dy\         /d'y\  x^     /d'y 


x^ 


3/       „  I  ■!"•••    > 

where  %,  (dy/dx)^,  (d^y/dx^)^,  etc.,  denote  the  values  which  y 
and  the  successive  derivatives  take  when  x  is  made  o.  "Differen- 
tiating equation  (2)  twice  in  succession,  and  then  placing  x=o, 
there  are  found 

yo=  +  io,         +10,  +iot,  —loi, 

(dy/dx)^  =-0.1125,  -0.2625,  +0.1875-0.0750^,  +0.1875+0.0750?*, 
(d^y/dx^)f^=  —0.002,0,  +0.0030,  —  0.0000+ 0.0039?,  —0.0000— 0.0039  i 

in  which  i  represents  the  square  root  of  negative  unity.  Sub- 
stituting each  set  of  corresponding  values  in  Maclaurin's  formula 
and  then  placing  x  =  i,  there  result 

;yi= +9.886,  >'3=o.i875+9.927i, 

^2= -10.261,  %=o.i875-9.927«, 

which  are  the  roots  of  (i),  all  correct  to  the  last  decimal. 


SOLUTIONS    BY    MACLAURIn's    FORMULA.  35 

This  method  may  be  readily  apphed  to  the  trinomial  equation 
y^  —  nay^''''  —  b=o.  When  x  is  inserted  in  the  second  term,  the 
series  obtained  is 

/    i\  l-ik 

+  \b")        (i-4^+w)(i-4^+2w)(i-4^+3w)aV4!  +  .  . . 

and  each  of  the  roots  is  hence  expressed  in  an  infinite  series, 
I 

since  b^  has  n  values.  This  series  is  convergent  when  a"  is 
numerically  less  than  k~^{n  —  kY~'^b^,  and  for  this  case  the  roots 
can  be  computed.  Now  the  condition  a^  =  k~^{n  —  kY~''^b^  is 
that  of  equal  roots  in  the  trinomial  equation;  hence  for  the  cubic 
equation  the  above  series  is  applicable  when  one  root  is  real  and 
the  others  imaginary,  while  for  the  quartic  equation  it  is  applicable 
when  two  roots  are  real  and  two  imaginary.  For  the  irreducible 
case  in  cubics  and  quartics  the  above  series  does  not  converge 
and  the  roots  cannot  be  computed  from  it ;  this  case  is  treated 
on  the  next  page  by  inserting  x  in  other  terms.  This  series  is  the 
same  as  that  derived  for  trinomial  equations  by  McClintock's 
method  of  enlargement  (Art.  14), 

Asa  special' case  take  the  quintic  equation  y^  —  '^ay  — 1=0,  in 
which  the  value  of  w  is  5,  that  of  k  is  4,  and  those  of  b'^  are  the 
five  imaginary  roots  of  unity  (Art.  15).  When  a  is  less  than 
4~^,  or  a  less  than  about  0.33,  the  above  series  applies,  and  if  s 
designates  one  of  the  imaginary  fifth  roots  of  unity  (Art.  15), 
the  five  roots  of  the  equation  are 


)'i=  I  +  a  -  a2  +  ^3  -  -Va'  +  -'/a'  -  ^^a'  +  ^^a^  -. 


•  > 


36  THE    SOLUTION   OF    EQUATIONS. 

For  example,  let  a=o.i,  or  y^  —  ^y  — 1=0;    then  the  value  of  y^ 
is  found  to  be  +1.09097,  while  the  other  roots  are 

}'2=+ 0.2  3649  + 1. 01470/,     >'3= -0.781975 +0.48372/, 
3-4=  +0.23649 -1. 01470/,     yi=  -0.781975-0.48372/, 

which  are  correct  in  the  fifth  decimal  place. 

For  the  case  w^here  a^  is  greater  than  k~^'(n  —  k)^~'^b^  in  the 
trinomial  equation  y^  —  nay^~'^  —  b=o,  the  roots  may  be  obtained 
by  inserting  x  in  other  terms  than  the  second.  To  illustrate  the 
method  by  the  quintic  ;y^  — 5a}'  — i  =0,  let  x  be  placed  in  the 
last  term,  giving  y^—<iay  —  x  =  o;  obtaining  the  derivatives  and 
making  n  =  o,  there  is  found  a  series  giving  four  of  the  roots, 
since  (5^)^  in  this  series  has  four  values.  Again,  placing  x  in 
the  first  term  the  equation  is  ^7^  — 5a;y  — i  =0;  and  applying  the 
method,  there  is  found  a  series  which  gives  the  other  root.  It 
may  also  be  shown  that  these  series  are  convergent  when  a^  is 
numerically  greater  than  4"^  When  a^  =  4~*  the  quintic  has  two 
equal  roots  and  the  series  do  not  apply,  but  in  this  case  the  equal 
roots  are  itadily  found  (Art.  5)  and  after  their  removal  the  other 
three  roots  are  found  by  the  solution  of  a  cubic  equation. 

When  this  method  is  applied  to  an  algebraic  equation  of  the 
11^^  degree  which  contains  more  terms  than  three,  there  may  be 
obtained  several  series  by  inserting  x  in  different  terms,  and  the 
series  desired  are  those  which  are  convergent.  A  general  rule 
for  selecting  the  terms  which  are  to  contain  x  is  given  by  Lam- 
bert, and  he  applies  the  method  to  the  solution  of  the  quintic 
equation y^  —  ioy^+6y  + 1  =0.  First,  writing y^  —  ioy^+6xy  +:x:  =  0, 
the  values  of  y^  are  +3.167  and  —3.167,  those  of  (dy/dx)^  are 
—  1. 00  and  +0.090,  and  those  of  {d'^y/dx'^)  are  —0.016  and 
+0.016;  inserting  these  in  Maclaurin's  formula  there  are  found 
;yj=+3.o5  and  y^^—^-o^)-  Secondly,  writing  xy^  —  ioy^+6y  + 
x  =  o,  a  series  results  which  gives  ^'3= +0.87  and  ;y4=— 0.69. 
Lastly,  writing  xy^^  —  ioxy^  +  6y  +  i,  there  is  found  ;y5=— 0.17. 

This   method   may   likewise   be  used  for  computing    one  of 
the  roots  of   a   transcendental   equation,  provided   the  resulting 


SYMMETRIC    FUNCTIONS    OF    ROOTS.  37 

series  is  convergent.  For  example,  take  2)'+log  )'  —  io  000=0. 
Writing  2}' +x  log  j  — 10  000=0,  there  are  found  the  values 
yo=+S  ooO'  {dy/dx)^=  -^  log  3^0,  and  (d"y/dx\=  +0.0001  log  y^. 
When  the  logarithm  is  in  the  common  system  the  root  isy  =  4998. 1 5 ; 
when  it  is  in  the  Naperian  system  the  root  is  )' =4995,74. 

Prob.  24.  Compute  the  roots  of  :x:^— 2:^—2  =  0  by  the  above  method 
and  also  by  that  of  Art.  9. 

Prob.  25.  The  equation  y^— 11  727y+4o  385  =  0  occurs  in  a  paper 
on  the  precession  of  a  viscous  spheroid  by  G.  H.  Darwin  in  Philosoph- 
ical Transactions  of  the  Royal  Society,  1879,  Part  ii,  p.  508.  Com- 
pute the  four  roots  to  five  significant  figures. 

Art.  17.     Symmetric  Functions  of  Roots. 

The  coefficients  of  an  algebraic  equation  are  the  simplest 
symmetric  functions  of  its  roots.     Let  the  equation  be 

x'^  —  ax'^~^+bx"'~^  —  cx^~^+dx^~*  —  ...=o,      .     .     (i) 

and  let  x,^,  X2,  x^,  .  .  .  be  its  n  roots.     Then 

CI       jC^  ~\~  JC2    I   ^3    1    .   •   •   J  U  ^^^ OC-tOCo  ~T~ ^2    3    i" ^z    A  "*   •  •  •  ) 

and  the  last  term  is  ^x^x^x^ .  .  .  Xn.  All  symmmetric  functions  of 
the  roots  may  be  expressed  in  terms  of  the  coefficients. 

The  sums  of  the  powers  of  the  roots  are  important  symmetric 
functions.  Let  Sm  represent  Xi^+X2^-\-x^^+.  .  .  ;  then  when 
m  is  equal  to  or  less  than  n,  the  following  ^e  the  Newtonian 
expressions  for  the  sums  of  the  powers  of  the  roots : 

5i  =  a,  82  =  0'^  — 2h,  S3  =  a^  —  ^ab+^c, 

Si  =  a*  —  4a^b+4ac  +  2b'^  —  4d,         .... 

Let  ±1  represent  the  coefficient  of  the  (m  +  iY^  term  in  the 
general  equation  (i),  this  being  +  when  ni  is  even  and  —  when 
m  is  odd.  Then  the  following  general  formulas  furnish  values 
of  5m  for  all  cases : 

Sm-CI'Sm-i+bSm-2-cSm-3+'  ■  -  ±^1    =0,  W<», 

<in+m      ^«Jn+m— 1    I   t'On+TO— 2      •  •  •   i''Jwi=0)  mT^fl, 


38  THE   SOLUTION   OF   EQUATIONS. 

For  example,  take  :v;^  -  2:v  -  2  =  o,  for  which  a  =  o,  Z>  =  -  2 , 
c=+2;  then  from  the  first  formula  8^=0,  5^  =  4,  ^3  =  6,  and 
from  the  second  formula  ^'4  =  8,  6'5  =  20,  56  =  28,  etc. 

Other  important  symmetric  functions  of  the  roots  are  the 
sums  of  the  squares  of  the  terms  in  the  above  expressions  for 
the  coefficients  b,  c,  d,  etc.     Let  these  be  called  B,  C,  D,  etc.,  or 

72   __  /y*    2/y*     2   _|_  /y»     2^     2    _|_  f*  /y*     2/yt     2/y.     2       1^  /y*     2/y*     2/v«     2       I 

xy  — ~ (A/j   *A'2     n^ *^2       3       *^ '    •   •   >         ^'^  —  *^1    *^2       3     ~f~      2       3       4       **  •   •   •  > 

and  let  it  be  required  to  find  the  values  of  B,  C,  D,  etc.,  in  terms 
of  a,  h,  c,  etc.     For  this  purpose  let  (i)  be  vv^ritten 

and  let  both  members  be  squared  and  the  resulting  equation  be 
reduced  to  the  form 

yn-Ayn-'^^By^-^-Cy'-^+Dy''-*-.  .  .==0,    .     .     (2) 

in  which  y  represents  x'^.  This  equation  has  n  roots  x^^,  x^, 
x^^  .  .  .  ;  hence  the  value  of  A  is  x^ -^x^ -\-x^ ^.  .  .  ,  and  the 
values  of  B  and  C  are  the  symmetric  functions  above  written. 
The  algebraic  work  shows  that 

A=a^  —  2b,     B  =  b'  —  2ac+ad,     C  =  c^  —  2bd  +  2ae  —  2/,    ...    , 

and  thus  in  general  any  coefficient  in  (2)  is  obtained  from  those 
in  (i)  by  the  following  rule:  the  coefficient  of  y"^  in  (2)  is  found 
by  taking  the  square  of  the  coefficient  of  x^  in  (i)  together  with 
twice  the  products  of  the  coefficients  of  the  terms  equally  re- 
moved from  it  to  right  and  left,  these  products  being  alternately 
negative  and  positive. 

An  equation  whose  roots  are  the  squares  of  those  of  (2)  may 
be  obtained  by  a  similar  process,  the  equation  being 

z^-^i2^-^+J5i2"-2-CiZ'^-'+Z)z^-''-.  .  .=0,  .     .     (3) 

in  which  A^,  B^,  Cj,  .  .  .  are  computed  from  A,  B,  C,  in  the  same 
manner  that  A,B,C,  .  .  .'were  computed  from  (i).  For  example, 
take  the  equation  :v^ +3^* +6=0;    the  equation  whose  roots  are 


LOGARITHMIC    SOLUTIONS.  39 

squares  of  those  of  the  given  equation  is  y''  +gy*  +  2,6y^  +  ^6=0, 
and  that  whose  roots  are  the  fourth  powers  of  those  of  the  given 
equation  is  z^+812^  — 6482^  +  19442^  — 25922  +  1296=0. 

Prob.  26.  Find  an  equation  the  roots  of  which  are  the  fourth  powers 
of  the  roots  of  x^+x+ 10=0. 

Prob.  27.  For  the  cubic  equation  x^—ax^+bx—c=o  show  that  the 
value  of  Xj^x^^+x^^x^^+x/xj^  is  b^—^abc+^c^. 

Prob.  28.  For  the  quartic  equation  x* — ax^ -\-  bx^ — cx+  d=  o  show 
that  the  value  of  S^  is  a^—  <,a%—  ^ab^+  ^a'^c—  ^ad—  ^bc. 

Art.  18.    Logarithmic  Solutions. 

A  logarithmic  method  for  the  solution  of  algebraic  equations 
with  numerical  coefficients  was  published  by  Graff e  in  1837  and 
exemplified  by  Encke  in  1841.*  The  method  involves  the  forma- 
tion of  an  equation  whose  roots  are  high  powers  of  the  roots 
of  the  given  equation;  to  do  this  an  equation  is  first  derived, 
.  by  help  of  the  principles  in  Art.  1 7,  whose  roots  are  the  squares 
of  those  of  the  given  equation,  then  one  whose  roots  are  the  squares 
of  those  of  the  second  equations  or  the  fourth  powers  of  those 
of  the  given  equation,  and  so  on.  With  the  use  of  addition  and 
subtraction  logarithms,  the  greater  part  of  the  numerical  work 
may  be  made  logarithmic.  The  method  is  of  especial  value 
when  all  the  roots  of  the  given  equation  are  real  and  unequal. 

To  illustrate  the  theory  of  the  method,  let  p,  q,  r,  s,  etc.,  denote 
the  roots,  each  of  which  is  supposed  to  be  a  real  negative  number; 
let  [p]  denote  p  +  q  +  r  +  .  .  .  ,  [pq]  denote  pq+qr+rs+.  .  .  ,  and 
so  on.     Then  the  general  algebraic  equation  may  be  written 

x''-[p]x"-'^+[pq]x''-^-[pqr]x''-^  +  [pqrs]x''-^-.  .  .  ,  (i) 

and  the  equation  whose  roots  are  p^,  ^,  r^,  .  .  .  is,  by  Art.  1 7, 

yn_[p2-^yn-l^^p2q2-jyn-2_^p2g2^2-^yn-S_^^p2q2j.2^2'^yn-i_^   .   .   , 

in  which  [p^]  denotes  p^  +  q^+r^  + . . . ,  [p-q^]  denotes  p^q^  +  q^r^ + . . . , 
*  Crelle's  Journal  fur  Mathematik,  1841,  Vol.  XXII,  pp.  193-248. 


40  THE   SOLUTION    OF   EQUATIONS. 

and  SO  on.  From  this  equation  another  may  be  derived  having 
the  roots  p^,  q*,  r*,  .  .  .  ,  and  then  another  may  be  found  having 
the  roots  p^,  <f,  r^,  .  .  .  .  This  process  can  be  continued  until 
an  equation  is  derived  whose  roots  are  p"^,  q"*,  r"*,  .  .  .  ,  where  m 
is  a  power  of  2  sufficiently  high  for  the  subsequent  operations. 
This  equation  is 

2n_|-^m]2n-l^[-^m^m]2n-2_|-^m^m;.w]2n-3_^ 

Now  let  p  be  the  root  of  (i)  which  is  largest  in  numerical 
value,  q  the  next,  r  the  next,  and  so  on.  Then,  as  m  increases 
the  value  of  [p"^]  approaches  />"»,  that  of  [p'^q'^]  approaches  p"^q"^, 
that  of  [p'^q^r'^']  approaches  p'^q'^r'^,  and  so  on.  Hence  when 
m  is  large  [^"*]  is  an  approximation  to  the  value  of  />"»,  and 
[p^q^y[p'^'\  is  an  approximation  to  the  value  of  q'^.  Accordingly 
by  making  m  sufficiently  large,  the  values  of  p^,  q^",  r"*,  .  .  .  , 
and  hence  those  oi  p,  q,  r, .  .  .  ,  may  be  obtained  to  any  required 
degree  of  numerical  precision.  When  two  roots  are  nearly  equal 
numerically,  it  will  be  necessary  to  make  m  very  large;  when 
equal  roots  exist  they  should  be  removed  by  the  usual  method. 

To  illustrate  the  appHcation  of  the  method,  let  it  be  required 
to  find  the  roots  of  the  quintic  equation 

x^  +  iT^x*  — Si  x^  — ^4x^+464% —  iSi  =0. 

By  comparison  with  (i)  of  Art.  17  it  is  seen  that  fl=  —13,  6=  —81, 
•c=+34,  d= -\-464,  g= +181.  The  equation  whose  roots  are 
the  squares  of  those  of  the  given  quintic  is  now  found  from  (2) 
of  Art.  17,  by  computing^  =a^  —  2b  =  T,;^i,  B  =  b^  —  2ac  +  2d  =  S^'j^, 
C  =  c^^2bd  +  2ae  =  'ji6i8,  D=d^  —  2ce  =  202gSS,  £  =  6^=32761, 
and  then 

y^  ~33i>'* +8373)'^- 7i6i8>'2  +  202988>' -32761=0. 

Taking  the  logarithms  of  the  coefficients,  this  equation  may  be 
written 

y'  -  i2.si9^3)y*  +  (3-92288))'^  -  (4.85502)/  +  (5.30747):y 

-(4-51536)  =0, 


LOGARITHMIC    SOLUTIONS.  41 

in  which  the  coefficients  are  expressed  by  their  logarithms  in- 
closed in  parentheses.  The  logarithms  of  the  coefficients  for  the 
equation  whose  roots  are  the  fourth  powers  of  the  given  quintic 
are  now  found  by  the  use  of  addition  and  subtraction  logarithmic 
tables,  and  this  equation  is 

z^  -  (4.96762)3"  +  (7.36364)2'  -  (9.24342)2^  +  (10.56243)2 

-  (9.03072)  =0. 

Next  the  equation  whose  roots  are  the  eighth  powers  of  the 
roots  of  the  given  quintic  is  derived  from  the  preceding  one  in  a 
similar  manner  and  is  found  to  be 

•zt'^  — (9.9329o)'Zi'"  +  (i4.3i934)'Z£^'  — (i8.i4025)w^  +  (2i.i2363)'Z£^ 

—  (18.06144)  =0, 

and  then  the  equation  whose  roots  are  the  sixteenth  powers  of  the 
roots  of  the  given  quintic  is 

'i;^-(i9.8658o)'y*  + (28.29778)7;^ -(36.i3i3i)'y2  + (42.24726)2; 

-  (36.12288)  =0. 

It  is  now  observed  that  the  coefficients  of  the  second,  fourth, 
and  fifth  terms  in  the  equation  for  v  are  the  squares  of  those 
of  the  similar  terms  in  the  equation  for  w.  Hence  two  of  the 
roots  are  now  determined  as  follows: 

log/=  9.93290,  log />  =  1.24161,     /?  =  i7.443; 

log  f  =  18.06144  — 21.12363,     log  /  =  !. 61723,      /=0.4142. 

These  are  the  numerical  values  of  the  largest  and  smallest  roots 
of  the  given  quintic,  but  the  method  does  not  determine  whether 
they  are  positive  or  negative;  by  trial  in  the  given  quintic  it  will 
be  found  that  —17.443  and  +0.4142  are  roots.  To  obtain  the 
others,  the  process  must  be  continued  until  two  successive  equa- 
tions are  found  for  which  all  the  coefficients  in  the  second  are 
the  squares  of  those  in  the  first.  Since  in  this  case  two  roots  lie 
near  together,  the  process  does  not  terminate,  with  five-place 
logarithms,    until    the    512'''    powers   are    reached.      The    three 


42  THE    SOLUTION    OF   EQUATIONS. 

remaining  roots  are  thus  found  to  be  ?= +3.230,  f= +3.213, 
and  s=  —1. 4142. 

When  this  method  is  applied  to  an  algebraic  equation  which 
has  imaginary  roots,  this  fact  is  indicated  by  the  deviation  of 
signs  of  the  terms  in  the  power  equations  from  the  form  as  given 
in  (2)  of  Art.  17;  that  is,  these  signs  are  not  alternately  positive 
and  negative.  As  an  example  of  such  a  case  Encke  applies  the 
process  to  the  equation 

X''  —  2X^  —  2,x^  +  /^X^  —  K^X  +6  =0, 
and  deduces  for  the  equation  of  the  256'''  powers  of  the  roots 

t;^- (74.95884)2^" +  (122.81202)2;^ +  (151.32153)7/^-1^179.58882)1^^ 
—  (190.99129)1;^  —  (195.21132)1'  — (199.20704)  =0. 

Here  it  is  seen  that  the  coefficients  of  v*  and  v  have  signs 
opposite  to  those  of  the  normal  form,  and  hence  two  pairs  of 
imaginary  roots  are  indicated.  The  real  roots  of  the  given  equa- 
tion are  then  determined  as  follows: 

log  Xi^^"  =  74.95884,  log  rTi  =0.29281,    :x:i= —1.9625, 

log;r2"®  =  i22.8i202—  74.95886,  log  :j£;2= 0.18693,  ^2= +i-5379> 
log  r)£:6"*'  =  i90.99i29  — 179.58882,    log  ^6  =  0.04454,   :V6= +1.1080, 

while  the  logarithms  of  the  moduli  of  the  imaginary  pairs  may 
be  obtained  by  taking  the  difference  of  the  logarithms  of  v^  and 
v^  and  that  of  v^  and  i'",  and  dividing  each  by  512.  It  is  then 
not  difficult  to  show  that  the  two  quadratic  equations 

^^— o.6o92ix  +  i.o7668  =  o,         jc^  + 1.29263^^1.66642  =0, 

furnish  the  imaginary  roots  of  the  given  equation  of  the  seventh 
degree. 

Prob.  29.  Compute  the  roots  of  x^—iox^+6x+i  =  o. 

Prob.  30.  How  many  real  roots  has  the  equation  x''+;}x*+6  =  o? 
Can  they  be  advantageously  computed  by  the  above  method  ?  What 
is  the  best  method  for  finding  the  roots  to  four  decimal  places  ? 


INFINITE    EQUATIONS.  43 


Art.  19.    Infinite  Equations. 

An  infinite  series  containing  ascending  powers  of  x  may 
be  equated  to  zero  and  be  called  an  infinite  equation.  For 
example,  consider  the  equation 

A*"  /y»D  /y»V  iy»9 

fc^  •A'  tA'  vv 

3!      5!      7!      9! 

in  which  the  first  member  is  the  expansion  of  sin  :r;  this  equa- 
tion has  the  roots  o,  tt,  27z,  t,7z,  etc.,  since  these  are  the  values 
which  satisfy  the  equation  sin  x=o.     Again, 

^^S  ^»4  /Y*^  ^*o 

%\'  vV  fcV  lA' 

2!     4!     6!     8! 
is  the  same  as  cosh  x=o,  and  hence  its  roots  are  ^Tti,  ^izi,  etc. 

The  series  known  as  Bessel's  first  function  when  equated  to 
zero  furnishes  an  infinite  equation  whose  roots  are  of  interest 
in  the  theory  of  heat  * ;  this  equation  is 

^  ~  2"' "^  2^  ~  2^T5^62"  "^  ^^T^."6V82  ~  •  •  •  ^  °' 

and  it  has  an  infinite  number  of  real  positive  roots,  the  smallest 
of  which  is  2.4048.  The  roots  of  equations  of  this  kind  may 
be  computed  by  tentative  methods,  and  when  they  are  approxi- 
mately known  Newton's  rule  (Art.  4)  may  be  used  to  obtain 
more  precise  values. 

As  an  example  take  another  equation  which  also  occurs  in 
the  theory  of  heat,  namely, 

/yZ  /y»3  ^y«4  At5 

vV  vV  *A^  wV 

It  is  plain  that  this  equation  can  have  no  negative  roots,  for  a 
negative  value  of  x  renders  all  the  terms  of  the  first  member 

*  Mathematical  Monograph,  No,  5,  pp.  23,  63. 


44  THE    SOLUTION    OF    EQUATIONS. 

positive.     Calling  the  first  member /"(:x;),  the  first  derivative  is 

1+2      22-3     2^-s^-4     2'-s^-4^-s 

By  trial  it  may  be  found  that  one  root  of  /(x)=o  lies  between 
1.44  and  1.45.  For  x  =  i.44,  /(x)  becomes  +0.002508  and/'(x) 
becomes  +0.4334.  Then  /(x)//''(:x;)  =0.0058,  and  accordingly" 
^^1  =  1.44 +0.0058  =  1.4458  is  one  of  the  roots.  Another  root  of 
this  equation  is  ^2  =  7.61 78.  In  general  equations  of  this  kind 
have  an  infinite  number  of  roots. 

The  term  infinite  is  sometimes  applied  to  an  algebraic  equa- 
tion having  an  infinite  root,  and  cases  of  this  kind  are  often 
stated  as  curious  mathematical  problems.  For  instance,  the 
solution  of  the  equation 

x  —  a  =  (x^  —  a\/x^  +  a^)^, 

when  made  by  squaring  each  member  twice,  gives  the  roots  x  =  fa 
and  x  =  o.  But  x=o  does  not  satisfy  the  equation  as  written, 
although  it  applies  if  the  sign  of  the  second  radical  be  changed. 
The  equation,  however,  may  be  put  in  the  form 


a      /  \a^     a* 

X      \        yx^    X* 


and  it  is  now  seen  that  :x;  =  00  is  one  of  its  roots.  The  false  value 
x  =  o  arises  from  the  circumstance  that  the  squaring  operations 
give  results  which  may  be  also  derived  from  equations  having 
signs  before  the  radicals  different  from  those  written  in  the  given 
equation. 

Prob.  31.  Differentiate  the  above  function  of  Bessel  and  equate 
the  derivative  to  zero.  Compute  two  of  the  roots  of  this  infinite 
equation. 

Prob.  32.  Find  the  roots  of  2\/x~-2=\/x—2,+\/x—i. 

Prob.  $s-  Consult  a  paper  by  Stem  in  Crelle's  Journal  fiir  Mathe- 
matik,  1841,  pp.  1-62,  and  explain  his  methods  of  solving  the  equations 
cos  ic  cosh  3C+ 1  =  0  and  (4~sx^)  sin  a;— 4xcosx=o. 


NOTES    AND    PROBLEMS.  45 


Art.  20.    Notes  and  Problems. 

The  algebraic  solutions  of  the  quadratic,  cubic,  and  quartic 
equations  are  valid  for  imaginary  coefficients  also.  In  general 
the  roots  of  such  equations  are  all  imaginary.  The  method  of 
McClintock  (Art.  14)  and  that  of  Lambert  (Art,  16)  may  also 
be  applied  to  the  expression  of  the  roots  of  these  equations  in 
infinite  series. 

As  an  illustration  take  the  equation  x^  —  ;^x+ 4.1=0.  By  any 
method  may  be  found  the  roots  x^^ —i,  X2=  —o.si-\-i.gT,6  and 
^3=  —o.5i  — 1.936;  two  of  the  roots  here  form  a  pair  in  which  the 
imaginary  part  is  the  same  for  both,  the  real  and  imaginary  parts 
of  the  complex  quantities  having  changed  places.  There  are, 
however,  many  equations  with  imaginary  and  complex  coefficients 
in  which  pairs  of  roots  do  not  occur. 

The  most  general  case  of  an  algebraic  equation  is  when  the 
coefficients  a,  b,  c,  .  .  .  in  (i)  of  Art.  17  are  complex  quantities 
of  the  form  m+ni,  p+qi,  ....  Such  equations  rarely,  if  ever, 
occur  in  physical  investigations,  but  the  general  methods  ex- 
plained in  the  preceding  pages  will  usually  suffice  for  their  solution, 
approximate  values  of  the  roots  being  first  obtained  by  trial  if 
necessary.  In  general  the  roots  of  such  equations  are  all  com- 
plex, although  conditions  between  m  and  n,  p  and  q,  etc.,  may 
be  introduced  which  will  render  real  one  or  more  of  the  roots. 

Prob.  34.  Show  that  the  equation  x—e^=o  has  many  pairs  of 
imaginary  roots  and  that  the  smallest  roots  are  o.3i8i±i.3372i. 

/y^  /y"  /y*4  ^O 

_  ^^      -  vv  lA"  fc-V  •A' 

Prob.  35.  Solve  — :-| — -,  +  —;  +  —;+.  .  .=  — i. 
^^  2!     3!     4!      5! 

Prob.  36.  Discuss  the  equation  :x;— tan:x:=o  and  show  that  its 
smallest  root  is  4.49341. 

Prob.  37.  Find  the  value  of  x  in  the  equation  e"^-}-i  =  o,  and  also 
that  in  the  equation  ei^*— i=o. 

Prob.  38.  Show  that  x^+(a+bi)x+c+di=o  has  one  real  and  one 
complex  root  when  the  coefficients  are  so  related  that  b^c+d^—abd=o. 


46  THE    SOLUTION    OF    EQUATIONS. 

Prob.  39.  When  and  by  whom  was  the  sign  of  equality  first  used  ? 
What  reason  was  given  as  to  the  propriety  of  its  use  for  this  purpose  ? 

Prob.  40.  There  is  a  conical  glass,  6  inches  deep,  and  the  diameter 
at  the  top  is  5  inches.  When  it  is  one-fifth  full  of  water,  a  sphere 
4  inches  in  diameter  is  put  into  the  glass.  What  part  of  the  vertical 
diameter  of  the  sphere  is  immersed  in  the  water? 

Prob.  41.  When  seven  ordinates  are  to  be  erected  upon  an  abscissa 
line  of  unit  length  in  order  to  determine  the  area  between  that  line 
and  a  curve,  their  distances  apart  in  order  to  give  the  most  advan- 
tageous result  are,  according  to  Gauss,  determined  by  the  equation 

Compute  the  roots  to  five  decimal  places  and  compare  them  with 
those  given  by  Gauss. 


INDEX. 


Abel's  discussion  of  quintic,  22. 
Algebraic  equations,  i,  2. 

solutions,  15-24. 
Approximation  of  roots,  3,  12,  49. 
rule,  6. 

Bessel's  function,  43. 
Binomial  equations,  16,  26,  31. 

Cardan's  formula,  17,  18,  28,  29. 

Catenary,  14. 

Cube  roots  of  unity,  32. 

Cubic  equations,  3,  17,  28. 

Cylinder,  floating,  13. 

De  Moivre's  quintic,  22,  26. 

theorem,  33. 
Derivative  equation,  9. 

Elliptic  solution  of  quintic,  23. 

Fifth  roots  of  unity,  16,  32. 

Graphic  solutions,  3. 
Graphs  of  equations,  9. 
Graff e's  method,  14- 

Horner's  process,  2,  12. 
Howe  truss  strut  problem,  21. 
Hudde's  method,  8,  12. 

Imaginary  coefficients,  45. 

roots,  II,  18,  20,  30,  34,  4 J 
Infinite  equations,  43. 

Lagrange's  resolvent,  15. 
Lambert's  method,  33. 
Literal  equations,  i,  10. 


Logarithmic  solutions,  39. 

Maclaurin's  formula,  34. 
McClintock's  quintic  discussion,  23. 
series  method,  29. 

Newton's  approximation  rule,  6.    I 
Numerical  equations,  i,  10. 

Powers  of  roots,  38,  40. 
Properties  of  equations,  11. 

Quadratic  equations,  16. 
Quartic  equations,  19,  20. 
Quintic  equations,  21,  36. 

Real  roots,  2,  3,  12,  40. 
Regula  falsi,  5. 
Removal  of  terms,  22 
Resolvent,  17. 
Root,  I. 

Roots  in  series,  27,  29,  31,  34. 
of  unity,  16,  31. 

Separation  of  roots,  8. 
Sixth  roots  of  unity,  17,  32. 
Sphere,  floating,  13,  25. 
Sturm's  theorem,  8,  12. 
Symmetric  functions,  37. 

Transcendental  equations,  2,  4,  13. 
Trigonometric  solutions,  24,  26. 
Trinomial  equations,  29,  35. 
Tschirnhausen's  transformation,  22. 

Vectors,  16,  ^^. 

Water-pipe  problem,  13. 


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Keep's  Cast  Iron Svo, 

Lanza's  Applied  Mechanics Svo, 

Lowe's  Paints  for  Steel  Structures 12mo, 

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*  Strength  of  Materials 12mo,  1  00 

Metcalf 's  Steel.     A  Manual  for  Steel-users 12mo,  2  00 

Morrison's  Highway  Engineering 8vo,  2  50 

Patten's  Practical  Treatise  on  Foundations 8vo,  5  00 

Rice's  Concrete  Block  Manufacture 8vo,  2  00 

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States 8vo.  2  50 

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■VoL  III.  Railroad  Engineer's  Field  Book.     (In  Preparation.) 

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1  aylor  s  Pnsmoidal  Formulae  and  Earthwork gvo 

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s. 


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*  ."  "  "  Abridged  Ed 8vo,' 

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French  and  Ives'  Stereotomy gvo' 

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Jamison's  Advanced  Mechanical  Drawing ■ gvo' 

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Elements  of  Geometry 8vo,  1  75 

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Wood's  Elements  of  Co-ordinate  Geometry 8vo, 

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MECHANICAL   ENGINEERING. 


MATERIALS   OF   ENGINEERING,  STEAM-ENGINES   AND    BOILERS. 


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Reid's  Course  in  Mechanical  Drawing 8vo 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. 8vo 

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Schwamb  and  Merrill's  Elements  of  Mechanism 8vo, 

Smith  (A.  W.)  and  Marx's  Machine  Design 8vo, 

Smith's  (O.)  Press-working  of  Metals 8vo, 

Sorel's  Carbureting  and  Combustion  in  Alcohol  Engines.      (Woodward  and 

Preston. ) Large  1 2mo, 

Stone's  Practical  Testing  of  Gas  and  Gas  Meters 8vo, 

Thurston's  Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics. 

12mo, 
Treatise  on  Friction  and  Lost  Work  in  Machinery  and  Mill  Work .  .  .  8vo, 

*  Tillson's  Complete  Automobile  Instructor 16mo, 

*  Titsworth's  Elements  of  Mechanical  Drawing Oblong  8vo, 

Warren's  Elements  of  Machine  Construction  and  Drawing 8vo, 

*  Waterbury's  Vest  Pocket  Hand-book  of  Mathematics  for  Engineers. 

2jX5|  inches,  mor. 

*  Enlarged  Edition,  Including  Tables mor. 

Weisbach's    Kinematics    and    the    Power    of    Transmission.      (Herrmann- 
Klein.) 8vo, 

Machinery  of  Transmission  and  Governors.      (Hermann — Klein.).  .8vo, 
Wood's  Turbines 8vo, 


MATERIALS   OF  ENGINEERING. 

*  Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo, 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering 8vo, 

Church's  Mechanics  of  Engineering 8vo, 

*  Greene's  Structural  Mechanics 8vo, 

*  Holley's  Lead  and  Zinc  Pigments Large  12mo 

Holley  and  Ladd's  Analysis  of  Mixed  Paints,  Color  Pigments,  and  Varnishes. 

Large  12mo, 
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Steels,  Steel-Making  Alloys  and  Graphite Large  12mo, 

Johnson's  (J.  B.)  Materials  of  Construction 8vo, 

Keep's  Cast  Iron 8vo, 

Lanza's  Applied  Mechanics 8vo, 

Maire's  Modern  Pigments  and  their  Vehicles 12mo, 

Maurer's  Technical  Mechanics 8vo, 

Merriman's  Mechanics  of  Materials 8vo, 

*  Strength  of  Materials 12mo, 

Metcalf 's  Steel.     A  Manual  for  Steel-users 12mo, 

Sabin's  Industrial  and  Artistic  Technology  of  Paint  and  Varnish 8vo, 

Smith's  ((A.  W.)  Materials  of  Machines 12mo, 

*  Smith's  (H.  E.)  Strength  of  Material 12mo, 

Thurston's  Materials  of  Engineering 3  vols.,  8vo, 

Part  I.     Non-metallic  Materials  of  Engineering 8vo, 

Part  II.     Iron  and  Steel 8vo, 

Part  III.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents 8vo, 

Wood's  (De  V.)  Elements  of  Analytical  Mechanics 8vo, 

Treatise  on    the    Resistance    of    Materials    and    an    Appendix    on    the 

Preservation  of  Timber 8vo,     2  00 

Wood's  (M.  P.)  Rustless  Coatings:    Corrosion  and  Electrolysis  of  Iron  and 

Steel 8vo,     4  00 


STEAM-ENGINES   AND   BOILERS. 

Berry's  Temperature-entropy  Diagram 12mo,     2  00 

Camot's  Reflections  on  the  Motive  Power  of  Heat.      (Thurston.) 12mo,      1   50 

Chase's  Art  of  Pattern  Making 12mo,     2  50 

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Creighton's  Steam-engine  and  other  Heat  Motors 8vo 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  ..  .  16mo,  mor 

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Goss's  Locomotive  Performance 8vo 

Hemenway's  Indicator  Practice  and  Steam-engine  Economy 12mo 

Hutton's  Heat  and  Heat-engines 8vo 

Mechanical  Engineering  of  Power  Plants 8vo 

Kent's  Steam  boiler  Econoiny 8vo 

Kneass's  Practice  and  Theory  of  the  Injector 8vo 

MacCord's  Slide-valves 8vo 

Meyer's  Modern  Locomotive  Construction 4to 

Mover's  Steam  Turbine 8vo 

Peabody's  Manual  of  the  Steam-engine  Indicator 12mo 

Tables  of  the  Properties  of  Steam  and  Other  Vapors  and  Temperature- 
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Valve-gears  for  Steam-engines Svo 

Peabody  and  Miller's  Steam-boilers Svo, 

Pupin's  Thermodynamics  of  Reversible  Cycles  in  Gases  and  Saturated  Vapors 

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Reagan's  Locomotives:  Simple.  Compound,  and  Electric.     New  Edition. 

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Snow's  Steam-boiler  Practice 8vo 

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Valve-gears Svo 

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cator  and  the  Prony  Brake Svo 

Handy  Tables Svo 

Manual  of  Steam-boilers,  their  Designs,  Construction,  and  Operation  Svo 

Manual  of   the  Steam-engine 2  vols.,   Svo 

Part  I.      History,  Structure,  and  Theory Svo 

Part  II.      Design,  Construction,  and  Operation Svo 

Wehrenfennig's  Analysis  and  Softening  of  Boiler  Feed-water.     (Patterson ,) 

Svo 

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Whitham's  Steam-engine  Design Svo 

Wood's  Thermodynamics,  Heat  Motors,  and  Refrigerating  Machines.  .  .8vo 


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MECHANICS    PURE   AND    APPLIED. 


Church's  Mechanics  of  Engineering Svo,  6  00 

Notes  and  Examples  in  Mechanics Svo,  2  00 

Dana's  Text-book  of  Elementary  Mechanics  for  Colleges  and  Schools  .12mo,  1  50 
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Vol.     I.      Kinematics Svo  3  50 

Vol.  II.     Statics Svo,  4  00 

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Vol.  II Small  4to,  10  00 

•*  Greene's  Structural  Mechanics Svo,  2  50 

Hartmann's  Elementary  Mechanics  for  Engineering  Students.      (In  Press.) 
James's  Kinematics  of  a  Point  and  the  Rational  Mechanics  of  a  Particle. 

Large  12mo,  2  00 

•*  Johnson's  (W.  W.)  Theoretical  Mechanics 12mo,  3  00 

L,anza's  Applied  Mechanics Svo,  7  50 

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*  Vol.  II,  Kinematics  and  Kinetics.  12mo,  1  50 

Maurer's  Technical  Mechanics Svo,  4  00 

*  Merriman's  Elements  of  Mechanics 12mo,  1  00 

Mechanics  of  Materials Svo,  5  00 

•*  Michie's  Elements  of  Analytical  Mechanics Svo,  4  00 

15 


Robinson's  Principles  of  Mechanism 8vo,   $3   00 

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Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,     3   00 

Wood's  Elements  of  Analytical  Mechanics •. 8vo,     3   00 

Principles  of  Elementary  Mechanics 12mo,      1   26 

MEDICAL. 

*  Abderhalden's  Physiological   Chemistry   in   Thirty  Lectures.     (Hall   and 

Defren.) 8vo, 

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Bolduan's  Immune  Sera 12mo, 

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Chapin's  The  Sources  and  Modes  of  Infection.      (In  Press.) 
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Ehrlich's  Collected  Studies  on  Immunity.      (Bolduan.) 8vo, 

*  Fischer's  Physiology  of  Alimentation Large  12mo, 

de  Fursac's  Manual  of  Psychiatry.      (Rosanoff  and  Collins.)..  .  .Large  12mo, 

Hammarsten's  Text-book  on  Physiological  Chemistry.      (Mandel.) 8vo, 

Jackson's  Directions  for  Laboratory  Work  in  Physiological  Chemistry.  .8vo, 

Lassar-Cohn's  Practical  Urinary  Analysis.      (Lorenz.) 12mo, 

Handel's  Hand-book  for  the  Bio-Chemical  Laboratory 12mo. 

*  Nelson's  Analysis  of  Drugs  and  Medicines 12mo. 

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Rostoski's  Serum  Diagnosis.      (Bolduan.) 12mo, 

Ruddiman's  Incompatibilities  in  Prescriptions 8vo, 

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Smith's  Lecture  Notes  on  Chemistry  for  Dental  Students 8vo, 

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METALLURGY. 

Betts's  Lead  Refining  by  Electrolysis 8vo,     4  00- 

Bolland's  Encyclopedia  of  Founding  and  Dictionary  of  Foundry  Terms  used 

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Iron  Founder 12mo, 

"  "  Supplement 12mo, 

Douglas's  Untechnical  Addresses  on  Technical  Subjects 12mo, 

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Johnson's    Rapid    Methods   for    the   Chemical    Analysis    of    Special    Steels, 

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Keep's  Cast  Iron 8vo, 

Le  Chatelier's  High-temperature  Measurements.      (Boudouard — Burgess.) 

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Metcalf's  Steel.      A  Manual  for  Steel-users 12mo, 

Minet's  Production  of  Aluminum  and  its  Industrial  Use.      (Waldo.).  .  12mo, 

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Smith's  Materials  of  Machines 12mo, 

Tate  and  Stone's  Foundry  Practice 12mo, 

Thurston's  Materials  of  Engineering.      In  Three  Parts 8vo, 

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page  9. 

Part  II.     Iron  and  Steel 8vo,     3   50 

Part  III.  A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents 8vo,     2  50. 

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Ulke's  Modern  Electrolytic  Copper  Refining 8vo.   $3  00 

West's  American  Foundry  Practice 12mo,     i    50 

Moulders'  Text  Book 12mo.     2  50 


MINERALOGY. 

Baskerville's  Chemical  Elements.      (In  Preparation.) 

*  Browning's  Introduction  to  the  Rarer  Elements 8vo, 

Brush's  Manual  of  Determinative  Mineralogy.      (Penfield.) 8vo, 

Butler's  Pocket  Hand-book  of  Minerals 16mo,  mor. 

Chester's  Catalogue  of  Minerals 8vo,  paper, 

Cloth, 

*  Crane's  Gold  and  Silver 8vo, 

Dana's  First  Appendix  to  Dana's  New  "System  of  Mineralogy".  .Large  8vo, 
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Large  8vo, 

Manual  of  Mineralogy  and  Petrography 12mo, 

Minerals  and  How  to  Study  Them 12mo, 

System  of  Mineralogy Large  8vo,  half  leather, 

Text-book  of  Mineralogy 8vo. 

Douglas's  Untechnical  Addresses  on  Technical  Subjects 12mo, 

Eakle's  Mineral  Tables 8vo, 

Eckel's  Stone  and  Clay  Products  Used  in  Engineering.      (In  Preparation.) 

Goesel's  Minerals  and  Metals:  A  Reference  Book 16mo,  mor.     3  00 

Groth's  The  Optical  Properties  of  Crystals.      (Jackson.)      (In  Press.) 

Groth's  Introduction  to  Chemical  Crystallography  (Marshall).  .....  .12mo,      1   25 

*  Hayes's  Handbook  for  Field  Geologists 16mo,  mor.      1   50 

Iddings's  Igneous  Rocks 8vo,     5  00 

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pipe  12mo,  60 

Merrill's  Non-metallic  Minerals:  Their  Occurrence  and  Uses 8vo,  4  00 

Stones  for  Building  and  Decoration 8vo,  5  00 

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States 8vo, 

*  Tillman's  Text-book  of  Important  Minerals  and  Rocks 8vo, 

Washington's  Manual  of  the  Chemical  Analysis  of  Rocks 8vo, 


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*  Crane's  Gold  and  Silver 8vo, 

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*  8vo,  mor. 

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Eissler's  Modern  High  Explosives 8vo, 

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Ihlseng's  Manual  of  Mining 8vo, 

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Peele's  Compressed  Air  Plant  for  Mines 8vo, 

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Wilson's  Hydraulic  and  Placer  Mining.     2d  edition,  rewritten 12mo, 

Treatise  on  Practical  and  Theoretical  Mine  Ventilation 12nio, 

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SANITARY    SCIENCE. 

Association  of  State  and  National  Food  and  Dairy  Departments,  Hartford 

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International  Congress  of  Geologists Large  Svo  1   50 

Ferrel's  Pooular  Treatise  on  the  Winds Svo,  4  00 

Fitzgerald's  Boston  Machinist ISmo,  1  00 

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Jacobs's  Betterment    Briefs.     A   Collection    of    Published    Papers    on    Or- 
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Metcalfe's  Cost  of  Manutactures,  and  the  Administration  of  Workshops.. 8 vo, 

Putnam's  Nautical  Charts Svo, 

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